Certain special Yo-kai in the game can “evolve”—that is, they can transform themselves into more powerful versions with new names. If a Yo-kai can evolve, it automatically makes the transformation once it hits a certain level, unless you choose to cancel it.
The exact level number required for evolution depends on the type of Yo-kai. Some Yo-kai evolve early on, while others need to be leveled up fairly high before they evolve. Typically, the later a Yo-kai is able to evolve, the more powerful its evolved form will be.
You can choose to cancel a Yo-kai’s evolution if you like, leaving it as is. If you cancel, you’ll get another chance to evolve that Yo-kai when it reaches its next level.