The most serious statement Kurt Vonnegut makes in "Harrison Bergeron" is found in the opening lines of the story:
The year was 2081, and everybody was finally equal. They weren't only equal before God and the law. They were equal every which way. Nobody was smarter than anybody else. Nobody was better looking than anybody else. Nobody was stronger or quicker than anybody else.'
This futuristic world has achieved what some people want more than anything else--for all people to be absolutely, unequivocally, and strictly equal. And it is a colorless, unproductive, ugly, and unsatisfying world.
Because everyone is not created with equal talents and gifts, something must be done to equalize the playing field for those who have something someone else does not have--which is everyone. Unfortunately, the level at which everyone must become equal is the lowest level. For example, those who have the ability to sing well must be handicapped until they sing no better than anyone else. Now, in this world, everyone is a terrible singer. The same is true in other areas of life: beautiful people have to wear masks, good thinkers have to have their thoughts disrupted, and those with pleasant speaking voices must apologize and try to make them unpleasant.
These handicaps make everyone clownish and unproductive, and the greater the natural gifts and abilities the more cartoonish his appearance. At the age of fourteen, Harrison Bergeron is seven feet tall, and he is clearly a young man who was blessed with more of everything.
Harrison's appearance was Halloween and hardware. Nobody had ever born heavier handicaps. He had outgrown hindrances faster than the H-G men could think them up. Instead of a little ear radio for a mental handicap, he wore a tremendous pair of earphones, and spectacles with thick wavy lenses. The spectacles were intended to make him not only half blind, but to give him whanging headaches besides.
Scrap metal was hung all over him. Ordinarily, there was a certain symmetry, a military neatness to the handicaps issued to strong people, but Harrison looked like a walking junkyard. In the race of life, Harrison carried three hundred pounds.
Harrison gets fed up with his hindrances and handicaps, and he wants to show everyone what he can do--which gets him immediately killed.
Vonnegut is obviously making the point that being equal is not a matter of being the same; that is only a kind of ridiculous surface equality which is impossible to achieve even if it were sensible or desirable. Instead, Vonnegut advocates celebrating individual differences (re-read the passages with the ballerina and Harrison and you will see it there).
Equal opportunity and equal rights are much different than strict and rigid equality in all things. This future world is not the world the author wants; it is the world he fears.