A quick note about "where" this idea of conservation of angular momentum.

We assume that no matter which direction our system is orientated, the laws of physics are the same so if we run an isolated experiment east to west, we would produce the same results if the system were oriented north to south. That is, we assume that the universe is isotropic. This is just an assumption based on the fact that we have not had any experimental evidence to dispute this. Using quantum mechanics, we can show that if a system remains unchanged by rotating either the system (an active transformation) or the coordinate system (a passive transformation) then angular momentum is conserved.

Noether's Principle states that for any symmetry in physics there is an associated conserved quantity. This is a specific example of this. (A conserved quantity is any quantity which does not change in an isolated system)

Let's hand-wave a little:

For example, let us consider a hydrogen atom in it's own little universe and the laws of physics are isotropic. The semi-classical electron orbits the proton happily with constant angular momentum. We then throw a switch which cases the electromagnetic interaction to change slightly in a certain direction (let's say Coulomb's constant increases - all other things constant, a higher Coulomb constant means a greater force) thus breaking the isotropy of the universe. Then the Coulomb force will increase, causing the potential energy to become more negative thus forcing (by conservation of energy) the kinetic energy of the electron to increase, thus increasing the speed, thus increasing the total angular momentum of the universe.

Note that we do not mean to imply that if it's just the force that is different in different directions that angular momentum is not conserved, but rather if the rules which describe the force are different in different directions then angular momentum cannot be conserved. Nor does this argument suggest that if space is isotropic that angular momentum is conserved, just that if space is anistropic then angular momentum is not conserved. A proof of this would case a bit of sophistication (usually shown in undergrad quantum mechanics).