He didn't--not in the short story, anyway. He was shot by Diana Moon Glampers, the handicapper general, after declaring himself emperor.
In the film version, Harrison kills himself because he knows that his death--shooting himself in the head on live television--is the only way to keep his message to the masses alive. He would rather die in this way that makes people sit up and take notice, and maybe take off thier handicaps, than exist in a world where people are taught to accept simply being controlled. He has tasted the sweetness of REAL art and REAL beauty, and he wants other to share that same experience.
He knows that if he doesn't make some drastic move, then everything will remain the same, and no one, except possibly the secret society that runs the country behind the scenes, will understand the importance of individuality and excellence. He sees that as a tragedy.
And his plan works--if you notice, it is his son who later watches the videos of Harrison sharing art with the world, and his son and friend take of thier handicapping headsets.