Why does Newton's Third Law actually work? My father explained to me how rockets work and he told me that Newton's Third Law of motion worked here. I asked him why it works and he didn't answer. I have wasted over a week thinking about this problem and now I am giving up.
Can anyone explain why Newton's Third Law works?
For reference, Newton's third law:

To every action there is always opposed an equal reaction: or the
mutual actions of two bodies upon each other are always equal, and
directed to contrary parts.

 A: No!  
No one can explain why it works. People can just explain how it works. But that's a completely different story. They may use more general principles (try asking this question in an advanced course of GR, mechanics or similar!), but such an answer won't contribute to the "why". If you are interested in math, you may look at the Noether theorem and symmetries for a theoretical explanation (which still does not fully answer the why).
To the why: The third law is simply an observation. The law holds in any non-relativistic frame (let's stay non-relativistic to keep things simpler, the principles are the same) situation. No one has yet found a counter-example, and you can't expect to find one either. More generally one could say that the third law is a different formulation of momentum conservation.
Why do there even exist physical laws? Why does is there a universe instead of nothing? Most probably, there is no answer to questions such as these. 
The goal of physics is to describe objects around us in their simplest terms, and predict their behaviour. Thus physics is great at converting why questions into how questions. Therefore, asking as many how questions as possible is great!
Further, whatever physical law one comes up with is subject to invalidation if contrary evidence is found. No one has managed to find evidence  contrary to Newton's third law yet - despite not having any idea of why it works. And hence it's true.
But do persist with your questioning and interest, that's the right way to learn science!
A: Newton's third law is a restatement of conservation of momentum or perhaps a direct consequence of the law of conservation of momentum.
We can understand it mathematically quite easily. The mathematical description given in this post is not rigorous but is sufficient to give the intuition required to understand the relationship.

Newton's second law: definition of force
Newton's second law defines force as the rate of change of  momentum. It can be mathematically expressed as:
$$\vec{F} = \frac{d\vec{p}}{dt}$$
where $p$ is momentum and $t$ is time.
$$$$
Newton's third law & conservation of momentum
Consider two isolated objects $A$ and $B$. Let object $A$ exert a force $\vec{F_{AB}}$ on object $B$.
As $\vec{F_{AB}}$ is the only force acting on object $B$, the momentum of object changes as follows:
$$\vec{F_{AB}} = \frac{d\vec{p_B}}{dt}$$
As momentum is conserved, if the momentum of object $B$ changes at a rate $\vec{F_{AB}}$, the momentum of something else must change at the rate of $-\vec{F_{AB}}$.
The only other object which can lose momentum in our case (of two isolated objects) is object $A$. Therefore, object $A$ changes its momentum at the rate of $-\vec{F_{AB}}$. 
Oh well, we can apply newton's second law again. If the momentum of an object is changing at the rate of $-\vec{F_{AB}}$, then a net force of $-\vec{F_{AB}}$ must be acting on it.
$$ \vec{F_{BA}} = -\vec{F_{AB}}$$
In a nutshell, this is newton's third law: for every action ($\vec{F_{AB}}$), there is an equal and opposite reaction ($-\vec{F_{AB}}$ or $\vec{F_{BA}}$).
A: Physics can not ultimately answer this kind of why question, but here are 2 way to think about it:
a simple way
What would it mean to push hard on something, if it did not push back equally hard? eg How hard can you push air with your hand?
a harder way
The 3-rd law is mathematically equivalent to (and so required for) the conservation of momentum. This naturally leads to the question Why is momentum conserved?. This can partially be answered by Noether's theorem and the symmetry of space. If you are 10 years old then understanding Noether's theorems will be a major challenge, but if you like 'why' questions they are one of the closest things physicists have to an answer. 
In this case it basically says that because the laws of physics as expressed in terms of Cartesian coordinates are the same no matter where you put the origin (the coordinate system and its origin is an arbitrary artificial invention) mathematically implies that momentum must be conserved. Following the logic of the argument justifying that requires understanding the Lagrangian formulation of physics.
A: Why do you want to know?
I'm not kidding.  That's actually an important question.  The answer really depends on what you intend to do with the information you are given.
Newton's laws are an empirical model.  Newton ran a bunch of studies on how things moved, and found a small set of rules which could be used to predict what would happen to, say, a baseball flying through the air.  The laws "work" because they are effective at predicting the universe.
When science justifies a statement such as "the rocket will go up," it does so using things that we assume are true.  Newton's laws have a tremendous track record working for other objects, so it is highly likely they will work for this rocket as well.
As it turns out, Newton's laws aren't actually fundamental laws of the universe.  When you learn about Relativity and Quantum Mechanics (QM), you will find that when you push nature to the extremes, Newton's laws aren't quite right.  However, they are an extraordinarily good approximation of what really happens.  So good that we often don't even take the time to justify using them unless we enter really strange environments (like the sub-atomic world where QM dominates).
Science is always built on top of the assumptions that we make, and it is always busily challenging those assumptions.  If you had the mathematical background, I could demonstrate how Newton's Third Law can be explained as an approximation of QM as the size of the object gets large.  However, in the end, you'd end up with a pile of mathematics and a burning question: "why does QMs work."  All you do there is replace one question with another.
So where does that leave you?  It depends on what you really want to know in the first place.  One approach would simply be to accept that scientists say that Newton's Third Law works, because it's been tested.  Another approach would be to learn a whole lot of extra math to learn why it works from a QM perspective.  That just kicks the can down the road a bit until you can really tackle questions about QM.
The third option would be to go test it yourself.  Science is built on scientists who didn't take the establishment's word at face value, went out, and proved it to themselves, right or wrong.  Design your own experiment which shows Newton's Third Law works.  Then go out there and try to come up with reasons it might not work.  Test them.  Most of the time, you'll find that the law holds up perfectly.  When it doesn't hold up, come back here with your experiment, and we can help you learn how to explain the results you saw.
That's science.  Science isn't about a classroom full of equations and homework assignments.  It's about scientists questioning everything about their world, and then systematically testing it using the scientific method!
A: On Why Why Is a Valid Thing to Ask in Physics
Although almost all of the answers and comments claim that there is no real room for explaining why something happens or holds in Physics, it is, of course, not true. You can explain why there is this or that law in Physics - and not just in a trivial way that says, "because we found it to be obeyed in experiments". That is the sociological reason behind why we wrote papers about it and why we included it in our textbooks. It doesn't really explain anything about the law itself as to why it works. As Feynman explains beautifully in this video (or as Weinberg has explained at many places in this fantastic book), the meaning of why is a bit tricky business in Physics because of the following: Ultimately, as mentioned in another answer, we will be explaining that a certain law works based on some fundamental law, which we don't really know why works. This raises the question as to whether we have really explained anything at all. The answer, at least to the Scientists, is most obviously, a yes. Because what we call the fundamental laws have a power of explaining all the other laws in a minimal way, and thus, they are the reason why all the other laws work.
What I mean can be visualized clearly in the following manner: Suppose you have $100$ different working laws for different things. Then, one nice day, you find that there is a single law that is not just a mathematically clever rewriting of those $100$ laws in a single line but is an actually different single law that reproduces all those $100$ laws and also produces some other $200$ laws (which you didn't know previously, but you found them to be true upon checking when you came to know that this new magical single law predicts them to be laws apart from the previously known $100$ laws). Any logically speaking person would call the newly found single law to be the reason for all those $300$ laws. We do have the pending work of explaining this new law (which we don't know is whether possible or not) but we definitely have explained the origin of those $100+200$ laws in a very scientific manner. When (and if) we somehow explain the origin of the new law then we will have explained why those $300$ laws work in an even better way but that doesn't prove that our previous explanation as to why those laws work wasn't an explanation at all. 
On the Question of Why Newton's Third Law Works
I will give an explanation as to why it works which can further be explained more fundamentally based on the most fundamental framework of nature that we know today, the Standard Model and General Relativity. But I will stay within the regime of fairly classical explanations as to why Newton's third law works. Have a look at the Edit 2 to this answer.
First of all, it doesn't work all the time! I think this is perhaps the most ignored and less celebrated feature of the breakdown of Newtonian laws. I will come to this point later. Let me first explain why it works when it works and it will make obvious when it shouldn't work. 
Let's consider a system of particles. There is something called momentum in this universe that remains conserved in all the processes that happen (the link is intended to clarify how to think of why certain quantities are defined the way they have been defined). And every particle has a certain well-defined momentum. Now, it has been found that the effect of an external influence on a particle, which we will call force, happens to be just how fast the momentum of a particle changes (some readers will know that this is the second law, but not to confuse it with a definition rather than a law, see the linked answer). Now, consider two particles interacting in some fashion. Since the total momentum needs to be constant throughout the time, one particle will gain the momentum at just the same rate as the second particle loses it. Since the force is simply this rate at which a particle changes its momentum, if the force on one of the particle is equal to some quantity, say $F$, then it will be $-F$ on the other particle because the changes in the momenta of these particles are just the opposite during any time interval. 
Now, the above line of reasoning will break down if we have something other than particles that can have momentum. It might be hard for the OP to visualize (but there is no other way), there is something called electromagnetic fields that are not made of particles (at least classically). They are just some things that exist in the universe apart from the particles. And we have found out that these fields can also have momentum. Thus, during an interaction with particles, they can carry away some of the momenta (in some sense) that these particles had. Newton's third law simply has no reason to be valid in this case and indeed, it not valid generically in the interactions of these fields with particles. 
Edit 1
Note that the fact that we know why certain laws work is a very well understood fact by physicists. (In some sense, consequently) We know when a certain reasoning or law doesn't really explain why the other law works even if the prior law reproduces the later. The major examples would be the cases in which even the later reproduces the first and the two are equivalent (or dual). Notice that in purely Newtonian Mechanics, one couldn't have identified the law of conservation of the momentum as a deeper law owing to the absence of both the 'Noether theorems' and 'the discovery that fields carry momentum'. In such a scenario, given the second law of Newton, both the conservation of the momentum and the third law of Newton are actually dual to each other and none of them explains why any of them holds. This makes it even clearer that there is some objective and definite meaning when we say something explains why something else holds. 
Edit 2
I have tried to describe the quantum mechanical origin of the conservation of the classical momentum (which then implies the third law of Newton under suitable situations as explained in this answer) in this answer.
A: Newton's Third Law works because the universe tries to be fair. If you push against something it makes no sense not for it to push back against you. Your hand pushes on the table, and the table pushes back just as hard against your hand. If it didn't push back, your hand would go straight through the table. The world would literally fall apart without that law. 
Things do go through each other. A swimmer goes through water; you walk through the air all the time. But in both cases, things are literally bouncing off you. Air molecules bounce off your body as you move, and water molecules bounce off your body as you swim. Both the air and the water push back against you just as hard as you push them. That's why you feel resistance when trying to walk against the wind, or why it's a lot harder to run in water the deeper it is. 
A: Newton's Third Law is a direct consequence of conservation of momentum, which essentially says that in an isolated system with no net force on it, momentum doesn't change. This means that if you change the momentum of one object, then another object's momentum must change in the opposite direction to preserve the total momentum. Forces cause changes in momentum, so every force must have an opposite reaction force.
In your rocket example, the rocket and its escaping fuel form an isolated system.* The escaping fuel was originally at rest, and now is moving very fast, so it's obvious its momentum has changed. So the rocket, being the only other object in the system, must also change its momentum to counter that of the fast-moving gas. Thus the rocket exerts force on the gas, which generates an equal and opposite force on the rocket.
But now you may ask:
Why is momentum conserved?
Conservation of momentum comes from an idea called Noether's Theorem, which states that conservation laws in general are a direct result of the symmetries of physical systems. In particular, conservation of momentum comes from translation symmetry. This means that the behavior of a system doesn't change if you select a new origin for your coordinates (put differently, the system's behavior depends only on the positions of its components relative to each other). This symmetry is found in all isolated systems with no net force on them because it is effectively a symmetry of space itself. Translation symmetry of a system is a consequence of homogeneity of space, which means that space is "the same everywhere" - the length of a rod doesn't change if you put it in a different place.
But now you may ask:
Why is space homogeneous?
In classical mechanics, this is one of the basic assumptions that allows us to do anything else. In reality, according to General Relativity, space actually isn't homogeneous - it curves in the presence of massive objects. But usually it's close enough to homogeneous that classical mechanics works well (gravity is pretty spectacularly weak, after all), and so the assumption of homogeneity holds.
*This is in the absence of gravity. If gravity is included, the isolated system would include any other gravitating masses.
A: I know i am too late  for this answer but I couldn't stop myself from answering :). Also I am going to use the contradiction method which we generally use in mathematics.
Let us assume that it doesn't work .
So , nothing around you follows Newton's third law.
Now , you take a spring and try to compress it by applying a force on that spring. Since there is no Newton's third law, someone can easily argue that you can compress that spring to a very high extent or  say to a point size  . Since you are not feeling any opposing force, it should be a piece of cake for you to compress the spring . But from daily experience , can you really compress it to a point size ? No !!! You do feel an outward push on you  which is opposing you from compressing that spring.
Also the moment you leave that spring, you no longer feel any push which means that what you were feeling was a result of your action.
This contradicts our assumption. So our assumption is wrong !!!
So it is truly said that when you apply a force on any object , the object also applies a force on you in the opposite direction and through thousands of experiments you can deduce that both the forces are equal in magnitude.
Hope it helps ☺️.
A: If you imagine pushing a spring between two fingers, the spring presses in each finger the same amount. Similarly with your arm, if you push against something you are simply straightening your arm, your arm pushes you away the same as the object.
The same is true of a rocket, When the fuel explodes it pushes out in all directions. The explosion is pushing the rocket up as much as it pushes the exhaust gasses down.
A: If two static forces act by balancing each other there is no resultant motion. There is no resultant force either. 
However if a dynamic force acts on any body its reaction comes only by motion (dynamics) of the object. We can say it is equivalent to force and another imagined  static D'Alembert's force in opposite direction. That is the basic difference between static and dynamic force and displacement actions. 
Another such example is a falling body. There is force as well as resulting  motion.
Except for the source of energy  a freely falling body (gravity)  and rocket ( expelled burning fuel ) work  dynamically the same way.
A: Let us take a very different approach from other answers, which are, in general, very good. I will only sketch; if someone is interested just drop a comment and I will try to find the time to elaborate (note: I'm not English mother tongue so language could be "a bit off"; bear with me).


*

*Our Science is actually a method we use to investigate our surroundings.

*This method is based on two independent, but interlocked disciplines:


*

*Experimental Science.

*Exact (often called "Theoretical") Science.


*Experimental tries to probe Nature with experiments, like a game where you pose specific questions to find out what is the concealed object.

*Exact Science, OTOH, has nothing to do with the real world; it is a mathematic construct, starting with a set of postulates and deriving properties via theorems (or equivalent mathematical derivations).

*The two run on parallel tracks and never really interfere directly.

*What keeps these two "parallel" and prevents them to diverge (too much) is, actually, men; scientists are often "switching train" or communicating with colleagues traveling on the "other train".

*People experimenting try to understand which "postulates" could explain observed behavior.

*People doing theoretic work try to "predict" what would happen is specific (often "strange") cases in the chosen framework and ask experimenters to verify.

*Division is almost never so sharp, but it's useful to understand background.

*Newton "laws" are nothing more and nothing less than postulates he used to construct his Exact Science: Classical Physics. They have exactly the same role of Euclid's "Five Postulates": they are the foundation on which to build a completely abstract construction.

*Euclid Geometry did a wonderful work to let us understand and manipulate our surroundings, but are not the only ones that can be used (see: bourbakism) and other non-Euclidean geometries are possible (and probably a better representation of our reality).

*Same can be said for Classical Physics, which is "good enough" to compute trajectories for the Moon and beyond, but has been "superseded" by more modern Theories (e.g.: Relativistic Mechanics and Quantum Mechanics).

*When something like this happens it's very rare the "Old Theory" is completely discredited, what normally happens is it becomes a "special case" of the new, more general, Theory; The previous set of postulates are thus "derived" and "explained" in terms of the new ones so the don't look like postulates anymore, they are "demonstrated" instead; "demonstration", however ultimately resides in another bunch of undemonstrable assertions.

*Ultimately the reason why certain set of postulates is chosen is because our experiments point in that direction; for Newton's Third law it's easy to see what happens if you throw a baseball ball while standing on skates (and manage not to fall). Deriving other postulates is a more complex matter and it proceeds essentially by trial-and-error (i.e.: you know a certain amount of "facts" are true and adjust your assumptions to get those results). For that reason is crucial to "predict" some non-obvious and not-yet-observed "fact" to validate your theory (i.e.: to go back to Experimental Science train).

