Generally, handicapping in sports is used to ensure that those who are of lesser ability can compete with someone who is better. For example, in golf, par for a good player might be three strokes, but for someone of lesser quality, par might be four strokes. Remember, the golfer with the lowest strokes under par wins.
Here's a hypothetical situation:
Two golfers tee off at a 400-foot hole. Golfer A has four shots to make par, while Golfer B has six shots. Golfer A makes par by hitting her ball in the hole after four shots. Golfer B goes one-under after hitting his ball in the hole after five shots. Therefore, Golfer B wins the hole by being one under par.
In "Harrison Bergeron," the purpose of the handicaps is to ensure that everyone was "equal every which way." This includes sports, which, as part of their stated purpose, is to prove the superiority of one team or player over another.
While there are no sports in the story, there is ballet. Some of these dancers are clearly more gifted as their handicaps are heavier than the handicaps belonging to other dancers. This ensures that every ballerina is equally bad and equally good.