This is colorful language. "They" are not really waving the pieces of paper. They are trying to show them to the speaker.
We "wave" our hands to say goodbye to someone, moving it back and forth in the air. We can also "wave" something we are holding. We "wave a flag." And, if we are holding a piece of paper in our hands, we could "wave" the piece of paper.
The idea is that the speaker is not paying much attention to what "they" are doing. The speaker has tuned out. The speaker is not really listening to the words that they are saying, only that they are "barking" (i.e. they seem angry, or indignant, or annoyed). The "pieces of paper" are some kind of business record or account or agreement. They are trying to show them to the speaker. The records probably show that there are unpaid debts. They are trying to pass the papers across the desk to the speaker. The speaker is angry. He doesn't want to know the details. He doesn't want to take the papers and look at them. So, instead of saying "they tried to hand me the business records," he's pretending that they are just waving the papers meaninglessly and that he doesn't understand that they are trying to hand them over. So he says "they waved pieces of paper."
He is describing the scene from his personal point of view, the way it seemed, not the way it would look to someone who wasn't angry.