To outgrow something is to become too big to fit in it, or to grow too mature for it. Most college students have outgrown their affection for stuffed animals. If they haven't, they might have a hard time making friends in the dorm.
As you grow older and bigger, you outgrow all kinds of things: clothing, your childhood bed, your ice skates. You'll also outgrow many things in the sense of being too old for them to be appropriate or as appealing as they were when you were younger. Just about everyone outgrows their favorite picture books and cartoons eventually. The earliest meaning of outgrow was "surpass in growth" or "grow faster than."