Snell's law relates angles of incidence and refraction. Did he know why refraction occur?

Probably not - the nitty-gritty detail of refraction is pretty confusing, and it requires better understanding of atomic physics than Snell would have had back in the 1600s. The fact that he could still come up with this law illustrates one of the fundamental principles of physics: that we can come up with useful rules to describe how the world works even if we don't completely understand what's going on.