Rank S is obtained by collecting all of the stars and rescuing all of the Leedmees in a single stage. Both single and multiplayer stages count towards this. This is easiest to do in A New Morning, the tutorial stages.
The only tricky one would be stage 3, where you lean over and they jump to your hand. Somehow I just kept kicking them or bringing my hand up too fast and hurling them up into the air.