It helps to remember that the "u" in Monster-Pups is pronounced like "ooooo" as that gives you a clue as to what this game's about. Yes, each player controls a monster that needs to use the toilet to do its business and drop a load, specifically a crystal that is placed underneath the monster.
Will the business happen, though? You place the monster on one of the toilets on the game board, use the stick end of the plunger to turn the wheel, and (maybe) a hole opens in the game board underneath the toilet. Listen for the plop! If you hear it, you can use the plunger to pick up one of your face-down tokens. With luck, you reveal a full roll of toilet paper to clean up after yourself. Whoever first finds three rolls wins!