George's handicaps affect him physically because he's forced to wear a bag filled with 47 pounds of birdshot, which tires him out, and he has a mental handicap placed in his ear that, while it affects his ability to think clearly, it hurts him physically.
George is forced to wear a bag attached to him because he is superior to the people who surround him when he is not at home. His wife tells him that "he's not competing" with anyone at home. The birdshot causes George to become physically exhausted, which prompts his wife into saying "Go on and rest the bag for a little while. I don't care if you're not equal to me for a while."
The mental handicap is attached to George because "his intelligence was way above normal." When the handicap in his ear goes off, George winces, he turns white and trembles, and his eyes turn red, and, again, according to his wife he looks "kind of wore out," which are all physical effects.
However, George does not seem to mind these effects because when his wife suggests that he removes some of the birdshot, he says no, because "If I tried to get away with it, then other people'd get away with it-and pretty soon we'd be right back to the dark ages again, with everybody competing against everybody else."
These handicaps reveal a kind of purpose in this quest for equality. The citizens who carry these burdens do so symbolically so that people who don't need handicaps like Hazel will not feel inferior.