Whenever you read speculative fiction, it's important to look at the story's themes and see how they are related to our world. Although Kurt Vonnegut's "Harrison Bergeron" was written about 50 years ago, many of the themes in the story are relevant today.
Perhaps the idea from "Harrison Bergeron" that is most relevant today is the idea of equality. In the story, society has chosen to blind itself to the fact that certain individuals have greater athletic, intellectual, and aesthetic abilities than others. In order to make this law, the government passed several amendments. The story makes it clear that most in the society agree with the laws. At one point in "Harrison Bergeron," Hazel Bergeron tells her husband, George, that he should make his handicap a little bit lighter to ease his burden. George snaps back, "If I tried to get away with it...then other people'd get away with it—and pretty soon we'd be right back to the dark ages again, with everybody competing against everybody else."
The idea George states is very evident in our world today. Whether we're looking at participation trophies for youth sports or letter grades in school, we are conditioned to believe anyone who makes us feel as though we are not special is a problem. Additionally, instead of praising extraordinary talent, we often feel the need to tear it down, with the exception of sports. As a teacher, I often see students who expect the highest grade for the least amount of work possible. All of these ideas are reflected in "Harrison Bergeron."
Now, some have used "Harrison Bergeron" to promote an agenda of anti-affirmative action, anti-civil rights, and anti-equal opportunity policies. I don't think this is Vonnegut's purpose in writing this story, though. I believe Vonnegut is clearly suggesting that we as a society are afraid of the exceptional and that we make ourselves feel better when there is no one who is exceptional.