Why do your cards have Pokémon and Trainer as types?

For flavor! They are direct replacements for Creature and Planeswalker, respectively.

How does evolution work?

Evolve - An activated ability that turns the card over, just like transform. Double-faced Pokémon cards have the same abilities on both sides (minus the evolve ability); only their power, toughness, and name are different. Pokémon with Evolve act just like any other double-faced cards. See this link for more info.

Evolve from - An alternate casting cost very similar to mutate from Ikoria, but it targets a specific Pokémon. They evolve into the Pokémon on top plus all non-evolve abilities from under it. Pokémon with Evolve from act just like any other merged permanents. See this link for more info.

Both "Evolve" and "Evolve from" count as evolution/evolving. Any triggered abilities that care about Pokémon evolving will trigger from both types of evolution. If a card affects what you can pay "to evolve Pokémon" it applies to both types of evolution.

Evolve is already an existing mechanic, what gives?

The word evolve is too perfect not to use, so we're using it anyway. The set is self-contained and not meant to be played with other sets, so hopefully it won't be too confusing. Every card with Evolve and Evolve from has reminder text as well.