The underlying message of "Harrison Bergeron," which does not lie very far from the surface, is that this type of equality is not worth the price you have to pay for it. Given the handicaps with which he has been weighed down, Harrison, understandably, rejects the concept of equality altogether and wants to create a society of vigorous individualism. Vonnegut's view, however, is subtler than that of his protagonist. He does not necessarily regard it as a bad idea for opportunities to be distributed equally (at least, there is nothing in the text to suggest this). However, he clearly sees that forcing equality of outcome on a society would reduce everyone to the lowest common denominator. The government handicaps the intelligent so that they are no longer exceptional in relation to others.
There is a related point about the confusion surrounding the word "equality." Apart from differentiating equality of opportunity from equality of outcome, there is a more basic mistake, emphasized in the first paragraph, of misreading "equal" to mean "the same." Equality before the law, for instance, does not require that everyone should have the same abilities or resources. It simply means that someone accused of a crime will be treated in the same way as anyone else, whether they happen to be rich or poor, handsome or ugly, clever or foolish.