In the dystopian society depicted in "Harrison Bergeron," it's considered unacceptable for anyone to be smarter than anyone else. Harrison's father, George, certainly falls into this category. A highly intelligent man, he's just the kind of person that the Handicapper-General, Diana Moon Glampers, considers dangerous to the ideal of absolute equality to which she's committed herself with such utter ruthlessness.
So, in order to drag George down to the general level of intellectual mediocrity in society, he's forced to wear handicaps to prevent him from thinking. George must wear an uncomfortable ear radio that regularly emits loud noises to make sure that he cannot concentrate on anything for long or develop any train of thought.
Smart people are a serious threat to the despotic regime of Diana Moon Glampers, and the last thing she wants is for them to be able to formulate ideas for getting rid of her and the totalitarian state she represents. It would so much better for the Handicapper-General if everyone were like Harrison's mother, Hazel, who doesn't have to wear any handicaps as she's not exactly the brightest tool in the shed. To a considerable extent, she exemplifies the ideal of intellectual mediocrity that Diana Moon Glampers wishes to impose on society.