The purpose of Vonnegut's story "Harrison Bergeron" is to convey the idea that

Identical abilities doesn't equate to equality.

In the story, everybody in the dystopian society was given impairments to 'level the playing field' so to speak. No one person was smarter or faster or different from anybody else. In doing this, it's supposed to be supporting the idea of a more 'fair' society; however, the supposed 'equality' that resulted from this was ultimately the impairment of the broader spectrum of society.