What does the story mean in your own words?

"Harrison Bergeron" depicts a dystopian future in which "equality" is misunderstood and enforced in an oppressive way.
In American history, the idea of equality that great American thinkers and philosophers have supported is that every American should have the same opportunities to succeed: "life, liberty, and the pursuit of happiness." If the government decides to limit any of those "unalienable rights" (as it says in the "Declaration of Independence"), then the people's freedoms have been taken away. The correct way to ensure equality without interfering with freedom is to encourage success and put infrastructure in place to help and motivate those who are born into situations which limit their opportunities.
In this story, the government has not done this. Their idea is to enforce equality by handicapping talented people and preventing those with less talent from bettering themselves. It is also a society obsessed and placated by mindless television shows. Note, that when Harrison tries to take over, he goes to a television station rather than a government building or executive committee. In this story, the government's strategy is "equality by limitation." In the ideal American society, it would/should be "equality by opportunity."