A pick-up game is when a bunch of folks show up at a field to play ball. It's usually announced -- like "every Tuesday and Thursday there will be a pick-up game at the community field" or something like that. Sometimes, it just happens. Then folks sort of line up and count off "one, two, one, two" like that until they've got two teams. Then they Play Ball!
It's all about playing the game. It could be your neighbor, your doctor, your teacher and/or your congressman who show up to play. Fun!
I live in a fairly rural part of the county (the world). Maybe this doesn't happen elsewhere.