This opening line is a succinct explanation of what kind of story this will be. It is in the future. Through the course of the story, the reader will learn how and why everyone is equal. This story is what is called a dystopia. Such tales often describe negative visions of what the future will be like. There is a long history of this kind of literature. Some other examples include Anthem, 1984, and Brave New World.
We learn that in this world of 2081, the government has a position called the Handicapper General. It is her job to find any citizens who have a physical or mental advantage over other average citizens. She then gives them mental or physical handicaps to make them equal to all other citizens. Since Harrison Bergeron is so much more mentally and physically advanced than all the other citizens, he is given the most difficult handicaps.
Note the word "finally" in the opening line as well. This indicates something readers of the story often miss. The word finally suggests that society has been striving for equality. This phrase of "striving for equality" (and others like it) sounds quite familiar. It has been used in the Civil Rights Movement, the Women's Liberation Movement, and any other crusade to grant equal rights to all citizens. Equal rights is one thing. In this story, Vonnegut suggests that if we are all equal in every way, the world would be an oppressive place to live. But the "finally" does serve as a warning that having a society of equality must not lead to limitations of freedoms. The progress in America is intended to grant equal rights, not equalizing limitations. This story serves as a reminder to that belief.