Is there an intuitive explanation for Newton's Third Law? Why does nature follow Newton's third law? I understand that when we apply a force there would be a reaction due to mass/inertia but why is it equal in magnitude? 
I know the derivation from Newton's second law but I'd like a more intuitive explanation for this. (I have taken the first and second law as true and as axioms)
 A: If the forces were not equal in magnitude then their net sum would be nonzero. You could make a perpetual motion machine by pressing different-sized springs together so one pushed harder on the other one and the assembly accelerated off into space.
Believe it or not, respectable people have seriously claimed to have done this with radiation pressure inside a microwave cavity resonator, thus creating a fuel-less spaceship drive. It took a lot of expert investigation to demonstrate that it did not work.
A: Probably the most intuitive derivation is that Newton's 3rd law is the law that leads to conservation of momentum in Newtonian physics. In the modern view, such conservation laws are related to symmetries and are seen as the fundamental thing from which other things are derived. In this case, the conservation of momentum is related to the fact that the laws of physics are the same at different locations.
So the intuitive "derivation" (leaving out the mathematical details) would be: the laws of physics are the same from place to place, this implies that momentum is conserved, which implies Newton's 3rd law for mechanical systems.
A: Simply put, Newton's 3rd Law is an observation that (at least in classical physics) has proved to be true through experiment. Several other equations and relations exist which are axiomatic (such as Schrodinger's equation) and are obtained simply through observation.
Intuitively, consider a universe which only contains a ball. Can the ball accelerate if it has nothing to push off of? If so, what is it moving relative to? A ball in its own universe can only move relative to itself - its velocity is zero; it has no acceleration.
Furthermore, consider a universe with a ball and a person. The person throws the ball, imparting an impulse which gives the ball velocity and momentum. The average speed of the universe relative to itself must be zero, so net momentum must be zero. To preserve a total momentum of zero, the person must receive an opposite momentum. This momentum is equal to mass times velocity. Force, mass times acceleration, is the time derivative of momentum. In order for the momenta to always be equal in magnitude and opposite in direction, the derivatives of the momentum should also be equal and opposite, ergo, Newton's 3rd law.
A: 
The third law of Newton ultimately just an approximation, that ends
  up replaced with the deeper ideas of momentum and angular momentum
  conservation.

Most of interactions in microscopic level is consequense of electromagnetic interaction via em field. 
This law is valid when the change in (angular) momentum of the field is negligible. For more details see this post.
So wee need consider third Newton law as consequence of momentum and angular momentum conservation.
