The society attempts to "guarantee" equality through the use of handicaps; these "handicaps" offset any qualities you have that might be extraordinary or talented. For example, Harrison's father, whose "intelligence was way above normal, had a little mental handicap radio in his ear" that rattled loud, distracting, painful noises into his ear that made him lose his train of thought. Because of this, it was hard for him to think coherently, with any intelligence. This, their society thought, helped "to keep people like George from taking unfair advantage of their brains." He also had to wear "forty-seven pounds of birdshot in canvas bag, which was padlocked around [his] neck", because he was taller and stronger than the average man.
This is the society's ingenious plan to keep people from being unequal. No one had to feel uglier, more stupid, or less talented than anyone else, ever. Unfortunately, the system does have its hiccups. Harrison, for one. He breaks onto the scene, equipped with an absurd amount of handicaps, and for a moment, tears them off and dances beautifully with a gorgeous ballerina (revealed to be gorgeous only after "he removed her mask"). This interruption doesn't last long, however; Harrison is shot down. And, Vonneget seems to be saying that the handicaps work quite well because his mother and father forget about it nearly right after it happens. It's a sad but thought-provoking tale about taking equality to dangerous levels.