To obtain a Dreamcatcher, you must combine a ring with a feather. Rings are obtained by getting high cumulative scores in games in Nightmare Rift, and feathers are obtained by doing the same in Daydream Oasis.
To use a Dreamcatcher, you must interact with Dali or Goya. It takes two Dreamcatchers to defeat each enemy, as the first one will break when interacting with them. Defeating one of the enemies gives you the Gem of their realm.