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.
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.
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.