"The girl likes to eat cheesecakes."
Translation:Του κοριτσιού του αρέσει να τρώει τσίζκεικ.
Because αρέσει means something like "be pleasing to" -- the subject is the pleasing object or action. (It's very similar to French "plaire" or Spanish "gustar", if you know those.)
So Το κορίτσι αρέσει would mean "The girl is pleasing (to someone); (someone) likes the girl".
But here, the pleasing thing is the eating cheesecakes and the girl is the one who is pleased by the eating -- this is marked with σε or with genitive.