Throughout the play, ballroom dancing is a metaphor for human relationships. Shmoop would be happy to explain it to you, but Sam does it best:
HALLY. Say you stumble or bump into somebody…do they take off any points?
[…] SAM. There's no collisions out there, Hally. Nobody trips or stumbles into anybody else. […] …like being in a dream about a world in which accidents don't happen.
HALLY. Jesus, Sam! That's beautiful!
SAM. Of course it is. That's what I've been trying to say to you all afternoon. And it's beautiful because that is what we want life to be like. But instead, like you said, Hally, we're bumping into each other all the time. […] The whole world is doing it all the time. […] People get hurt in all that bumping, and we're sick and tired of it now. (1373-1375; 1385-1389; 1390-1391; 1394-1408)
Hally gets it:
HALLY. You know, Sam, that's what the United Nations boils down to…a dancing school for politicians! (1433-1435)
Dancing's not only a metaphor for social harmony; it has another meaning for Sam and Willie. It's the only time when they can escape for a while from the apartheid system that treats them like non-persons. The descriptions of the dance and dancers have a dreamlike quality as Sam tells it. While they're dancing, they're free. The dance competition is for black South Africans; it's their own special event, one of their few chances for self-determination. The dancers are black and so are their judges.