How has this society evolved?
In the short story "Harrison Bergeron," society has "evolved" to make everyone more or less equal to everyone else. Because in the past equality had been so valued, the people in power thought it best to make some changes which would ensure that everybody would be equally represented in society. In this case, they took it to mean that nobody could be considered "better" than anyone else or be given an unfair advantage, even if the culprit were simply being born better-looking, or with a higher intelligence than someone else. In the story, their society thought it best to create a government agency which would be responsible for enforcing these "handicaps." The woman in charge of it is known as the "Handicapper General."
For example, if a person were considered highly intelligent and capable of advanced thought, they would be handicapped so as to bring them down to a more acceptable level of functioning. In the relationship between Harrison's parents, Hazel and George, George is the extremely intelligent one, and so he has to wear headphones which emit loud, obnoxious noises that inhibit his ability to form complete thoughts. This was done in order to bring him down to Hazel's level. She doesn't have to wear the headphones, because she is already of below-average intelligence.
When Hazel and George watch the ballet on TV, they can tell which ballerinas must have been the most beautiful, because they are forced to wear masks so that they will not be given preferential treatment over anyone else. Those who have very attractive voices must develop a stutter or wear a voice-changing machine to mask their beautiful cadence. Also, those people who are tall and strong must be weighed down by heavy objects so as not to award them any physical advantages over others. In this way, everyone is considered to be kept on an "equal" level. They considered this an "evolved" way of living.