OK this literally is quantum physics and the discovery was Nobel-prize material but it's not that hard to grasp so here's how it works :)
If you have a chemical reaction that requires a certain energy level, shining lots of light on it won't help much if each individual photon is below that energy level. Even if you have a really bright reddish light, each photon still only has a relatively low energy. On the other hand each UV photon has more energy than a red one, so they can add enough energy to cause a chemical reaction.
So basically the uv light is kicking the electrons up multiple levels in the "photoinitiators", the "photoinitiators" stabilize by dropping levels and giving off photons. These photons excite the "Monomers" and "Oligomers" between which electrons can freely flow (radicals) making it possible to form new bonds, creating longer chains of atoms (a polymer) /amateurexplenation
Not sure about the exact chemistry but I think there are actually two exposures, one to harden the shape after which the unused material is removed, and then another higher one to make it less sticky or "detackify" it.