Why was Diana Moon Glampers, the Handicapper General, so afraid of Harrison Bergeron breaking free of his handicaps?
Like any powerful person or group, Diana Moon Glompers is most afraid of things that threaten her power. In this case, Harrison Bergeron represents an existential threat to the empire of "equality" that Glompers has created because he will be able to break down her system. Harrison's immense intellect, coupled with massive physical strength, will be enough to undo a great deal of the harm that Glompers has caused.
Given the clear display of the dangers of forced equality, it is clear that Vonnegut also wants to show the fear that those with lesser talents have when confronted with the real deal. It is difficult for humans to be humble, particularly if they are in positions of power or influence as this tends to emphasize ego and make it more difficult for them to admit weakness. The character of Diana Moon Glompers effectively demonstrates this tendency on the part of humans.