In this classic science fiction story, Vonnegut tells us some things directly about the society of 2081 and leaves the reader to infer others. The first detail given is about the equality. "THE YEAR WAS 2081, and everybody was finally equal. They weren't only equal before God and the law. They were equal every which way."
The second detail we see is that this society engages in continual monitoring to ensure people are equal. That's where the "unceasing vigilance of agents of the United States Handicapper General" comes in.
The third detail is that this is still technically the United States, and that rule of law still exists. This is indicated in the line "All this equality was due to the 211th, 212th, and 213th Amendments to the Constitution." This is an example of Vonnegut implying more than he says. If there are that many amendments to the Constitution, it is unlikely the country is still the same.
As for how society treats someone who is not average, we don't have to infer that as the story tells us directly. If you are above average in any way, the handicappers adjust your abilities. For example, they disguise the beautiful and disrupt the thinking of the intelligent.