From a layman’s perspective, if something spins, you need to apply a force to it to stop spinning, otherwise it will just keep spinning. Hence, conservation of angular momentum. If you change the mass distribution, it will spin faster or slower, just like ice skaters (this is probably the most striking example), but the angular momentum is still conserved, regardless their linear momentum. There is no fundamental difference between this and what happens at the atomic scale.

The not layman’s answer is Noether’s theorem: the invariance of the laws of physics after a rotation of the frame of reference implies conservation of angular momentum.