How does this work? In the case of Covid, I understand (for some value of "understand") how they can make the cells create virus proteins, which then train the immune system without an actual infection to be ready for when the real virus strikes. But if someone already has cancer cells, wouldn't all the proteins to train against already be in the body in the first place?

A cancer vaccine would work by priming the immune system to target the proteins that is mutated in the tumor cells, and killing the cells producing them.

In fact, the body does this regularly. Cells frequently mutate to divide uncontrollably in the body, but are nearly always suppressed by the immune system. It is only when the tumor somehow manage to bypass the immune system or the immune system is sufficiently weakened that cancers can spread.