Most of the exaggerated portions of Kurt Vonnegut's "Harrison Bergeron" have to do with the title character and his physical characteristics and dramatic actions.
The first example of the exaggeration occurs when the news describes the escape when they say he's an athlete and underhandicapped so "should be regarded as extremely dangerous" (even though there is no mention of violence in his description).
The narrator then describes how Harrison was "exactly seven feet tall" and then describes his walking as "an earthquake" and how the studio shook when he stomped his foot.
The narrator uses several exaggerations in how Harrison removed the handicaps. He tore them off "like tissue paper" and he snapped the bar that held his head harness "like celery."
Then the narrator describes how, once the handicaps were removed, it "revealed a man that would have awed Thor, the god of thunder."
Finally, Harrison's actions once the handicaps were removed are greatly exaggerated. It's highly unlikely that Harrison lifted two musicians off their chairs and "waved them like batons." It's also unlikely that the laws "of gravity and the laws of motion" were abandoned. They probably didn't leap like "deer on the moon" and kiss the ceiling.
There are other examples of exaggeration, particularly when they discuss the ballerina who must be "extraordinarily beautiful, because the mask she wore was hideous. And it was easy to see that she was the strongest and most graceful of all the dancers, for her handicap bags were as big as those worn by two-hundred pound" had a voice that "was a warm, luminous, timeless melody."
Vonnegut's reason for including these exaggerations is most likely to show the possibility of the heights people can reach when they are not sinking to the lowest common denominator. He could also be showing the strength, power, beauty of an individual who attempts to stand out.