Conservation of angular momentum tells us?

Angular momentum will not change unless an external torque acts upon the system

The short answer would be that angular momentum is conserved, i.e. it cannot be created nor destroyed.

A more technical answer would be that there is a certain theorem in theoretical physics called Noether's theorem which shows that if a physical theory exhibits rotational invariance (i.e. the physics are the same even if you rotate the system) that angular momentum conservation is a result.

According to particle physics therefore the conservation of angular momentum seems to tell us that the Universe is invariant under rotations. This might seem strange, because surely rotating yourself changes how think look, but the physics involved remains the same.