The mother is talking about growth spurts, which occur during the teenage years.
Pediatrics A period of rapid growth in middle adolescence
The mother is saying that she expects the children to have a growth spurt soon, so she'd rather buy shoes that are a little too big than ones they will grow out of before they have been worn for very long. As you said, it's comedy. Any parent who has seen a child quickly grow out of an expensive pair of tennis shoes might chuckle at her remark.
You probably already learned about these in health class, but for the sake of completeness, you can read more about growth spurts on Wikipedia.