Evolution is the process of changing a Monster's form. In Monster Sanctuary, only a few Monsters can evolve, and evolution is not a direct upgrade to the base Monster but rather simply an alternative form. For example, Magmapillar is good at sustaining your team by applying shields, Barrier and Regeneration to the party, whereas Magmamoth is more focus on dealing direct damage through Critical Hits and Charge stacks.
You can evolve Monsters at the Tree of Evolution in the Ancient Woods using special items called catalysts. Each catalyst is only usable on a specific Monster, except for the Majestic Crown. Once a Monster has evolved, the process cannot be reverted, and the catalyst is consumed. A Monster will retain their Shift as well as their respective Shift Passives upon evolution. Unlike other Monsters, evolved Monsters (except King Blob in Randomizer) do not drop their eggs but instead drop their pre-evolved form's eggs.