You can unlock the Auto battle option for a particular stage by first earning an "A" or "S" rank on it. level, which is done by completing the level under certain conditions. The following four conditions will determine your rank. Complete all four to earn an "S" rank or three of them for an "A" rank.

Complete the level normally.
Complete the level solo.
Complete the level under a particular time limit.
Complete the level without taking a certain amount of damage.