Property does not require escrow to transfer. People who want to be careful use it, but it is not required.
The purpose is to have a third party make an exchange of the items submitted to escrow so neither gets the other's item(s) before the other. It is a matter of trust--or lack thereof. In most cases things work just as FlyingRon has said.
Here, I'm guessing the deal was made and the deed changed hands and was recorded without escrow. One party is claiming that the quitclaim or warranty deed, because of the legalese or somesuch saying for good and valuable consideration the deed was transferred, that payment must have been made.
Of course, that is just silly. While the one who deserves the money still deserves the money, it's just that he may have to use legal process to get it. That's why people usually use escrow. It is far cheaper and more certain than litigation.