What is Chern-Simons theory?

Not sure that I am saying anything that is not in nLab, but let me try a birds eye view. A quantum theory is "mostly specified" by an action, and the CS theory in $2+1$ dimensions with group $G$ has as action the Chern-Simons functional (comes from boundary terms of characteristic classes) on the space of connections on some $3$ manifold. There is a parameter, and you get a well-defined theory whenever the parameter is a root of unity. Because this theory requires no background metric or other geometry to define, any computations in the theory should in principle be topological invariants of the manifold (life gets complicated in the details, but pretty much all you have to add is this extra biframing info to make that correct). Since particles are roughly representations, a sequence of particles interacting and moving around each other in space (a $2$-manifold here) will as a movie trace out linked loops labeled by representations in spacetime (a $3$-manifold). The expectation value of this sequence of interactions (roughly the probability of occurrence) is the value of the Jones polynomial at that value of $q$ (Those who have tried to make the various normalizations of the parameters in the physics and math literature align have gone mad: don't try it at home! It is pretty mysterious why these values combine to a polynomial). The value of the partition function for an ordinary manifold (which technically should not have physical meaning as you are supposed to divide out by it, but it is telling you something about the time-evolution operator, which is constant because it is topologically invariant).

All of that should is because the natural way to build a theory from the action is the path integral, which is a nonrigorous heuristic. The mathematical response to this is, in this case, axiomatic TQFT. Heuristic reasoning argues that the basic building blocks of the theory should have certain properties that should uniquely specify them, and then you can explicitly construct such objects from, say, quantum groups (I do NOT know how to get the quantum groups themselves from the physics) and prove they satisfy the necessary properties. From these you can compute partition functions and expectations to your hearts delight.