When people disagree, there is tension "in the air" between them. "Clearing the air" refers to removing this tension. In your example, this is done through negotiations. This phrase can also mean "coming clean". For instance, let's say I accidentally scratched your car and didn't tell you about it for a month. You were mad that someone scratched your car and I was afraid to admit that I was the one who did it. Finally, I decide to "come clean" or "clear the air" by telling you the truth that I was the one who scratched your car.

"Clear the air" means to get everything out in the open - to tell each other everything, absolutely everything, that is relevant to the situation at hand and then resolve it.