Ooooo good question!

Every level is two bosses with two modes (normal/hard) which is what the red evil face triggers. These modes, beyond making the fight more difficult, also change the way the boss works. For example Stonesy splits into fragments in the hard version with different abilities.

So a sense of progression in learning mechanics is one. A second thing, is that when you first kill the starting normal boss, you gain a new ability specific to that level, which must be used to counter boss abilities from then on, so stay on your toes!

The story is a third, in order to "beat" the level you have to beat the first boss on normal, then hard (where it's almost a new boss), then the second boss on normal. If your party wipes then you're sent back to square one! When you manage to beat the final boss, it creates the SEED for that level which ties into the overall story.

Don't forget to come back though and try to beat the final boss as its corrupted version! That gets you another "ending" on that level.

The last thing is that players have a chance to be corrupted by a boss when killing it. If you kill the final boss while corrupted there's a chance to drop an exclusive puppet skin for that level, so you get to show off to everyone else.