Is there an explanation for action-reaction law? If a body $A$ exerts a force over body $B$, $B$ exerts a reaction force over $A$. Is there an explanation of why this happens?
 A: Yes, the explanation is the conservation of momentum. In Newtonian mechanics the third law produces conservation of momentum in mechanical systems.
Later on you will see cases (matter interacting with fields) where Newton’s 3rd law is violated in some sense, but in these cases the conservation of momentum still holds (the fields have momentum).
Conservation of momentum (and its associated spatial translation symmetry) have no explanation for why it is true. We have lots of solid experimental evidence that it is true, but no explanation why our universe behaves that way instead of some other way. This is what makes conservation/symmetry laws fundamental explanations. There are no further explanations in physics, just evidence that makes us believe this explanation.
A: We take it to be true because it captures what we see in experiments and observations. An example is the way a rocket works. It is mathematically related to more abstract physics, but that's not really an explanation.
A: Is there an explanation for action-reaction law?
Your question: If a body A exerts a force over body B, B exerts a reaction force over A. Is there an explanation of why this happens?
I believe that the correct answer: Nobody knows.
You have asked a truly deep question!
My answer: First of all I am not a Physics Expert. I was trained in nuclear physics while in the military, and my line of work required me, for 35 years, to look deep into the nature of things.
That being said, I will give you two answers.
Case 1: Contact forces: Two hard objects collide. A kind of short-cut explanation for what happens when two (or more) hard objects, like pool balls, collide.
Note: The following answer is not technically correct, in the strictest sense, but it should suffice for conveying the general idea: (I'll probably get my head handed to me for all this, but look at the bright side - when a real expert tears my answer apart - you WILL get the correct "state of the art" explanation. Which everyone will swear by... until they make "a discovery"!!!)
Okay,
Try to think of the collection of forces, created by the electrons around the molecules/atoms, of the two objects like little tiny, ultra efficient "springs". When the two objects collide, the little springs cause the individual collections of objects to "bounce" away from each other. That is to say, we claim that total energy is conserved because we don't measure down to the atom. We assume that all works according to theory, and that is what is taught in schools.
Why does that work? I say: The attractin of the little springs to each other, is greater than the attraction to the little springs in the second "object".
a little more info:
Observation shows us that the total (mass x velocity) (specifically (m x v^2)/2) of all of the objects under consideration, both before and after the collisions, will always sum up to the same quantity (assuming energy isn't lost to heat, noise, or physical deformation).
If this ability to show the same amount of kinetic energy, before and after a collision, is due to the efficient workings of the "little springs", then the equal and opposite reactions observed - are due to electric forces, generated magically, by the electrons. (can you tell that I am a cynic?)
Case 2: An astronaut in space pushes a bowling ball away from himself, and both he and the bowling ball move away from each other. What is going on there?
I believe they "move apart" because there is no such thing as an absolute "location" anywhere in the universe. That means that there is no place that you can call the "center", and therefore, all measurements are relative, and in fact, arbitrary! Over time, humans have slowly come to agree on using certain measuring systems (but I degress).
When the astronaut pushes away the bowling ball, it is possible to compute the kinetic energy before and after the push, and again, we observe that (m x v^2)/2 remains constant. Now, I don't really understand relativity, but, (fools rush in where angels fear to tread) I will venture a guess: I think it goes something like is:
energy causes local space-time to "relax" as the two objects separate, and that this is why they appear to be moving farther apart.
My logic (if there is any in relativity) goes like this:
If mass contracts space-time, then the local space-time contraction caused by the total mass of the two objects has to be greater than the local space-time contraction of each object individually. That means that the applied energy causes each object to take on unique motion vectors. Once that happens, they each distort space time less around the observer... or something like that. Hence... "motion". However, even after changes to the "system" (the bowling ball, and the astronaut), we can make calculations that satisfy our belief that "total energy is conserved".
The real answer as to why the conservation law is regarded with such reverence, is that no one has succeded in creating perpetual motion (except "God" - you see... the planets, and all the atoms display the very perpetual motion that all scientists claim cant exist... what's up with that?)
AS I SAID: I am not an expert, and while this opinion is imperfect, I will claim that any answer is imperfect. We can only make claims on what has been observed so far. Eh?
