The opening paragraph of Kurt Vonnegut's "Harrison Bergeron" quickly establishes the setting for 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. All this equality was due to the 211th, 212th, and 213th Amendments to the Constitution, and to the unceasing vigilance of agents of the United States Handicapper General.
These seven sentences give us a lot of information. First of all, the story is set in the United States but at a future date. In the roughly one hundred years which have elapsed since the story was written, nearly two hundred Amendments have been added to the Constitution (do the math and the percentage of increase is gigantic). We presume that most of them are connected to the goal of making everyone equal under the law, and it is the details of that which begin to make us uneasy.
When we read that every single person in the country is now "equal in every which way," we are nervous. When we read that no one is smarter than anyone else, our fear is that now everyone is dumb. When we read that no one is better-looking than anyone else, we are certain that now no one is good-looking (that everyone is now plain or, even worse, ugly). When everyone has the same level of strength and quickness, we know that everyone is now weaker and slower.
Just as frightening as these these results is the way they have been achieved. We learn more details as the story progresses, but what we know from the first paragraph is that this hoped-for and much-lauded equality is vigilantly maintained by government agents. We have the slightest hint of what is to come when we see the word "Handicapper," but it is just a suggestion. In any case, it does not bode well for the country when equality is maintained through handicapping.
"Handicapping" is setting someone at some sort of disadvantage for the benefit of someone else: society is maintained as "equal" because Handicappers see to it that disadvantage is distributed where needed to maintain an equality in all areas.