It's fairly straightforward. I give you a pig and say you owe me a pig. You are in debt. Eventually somebody notes down I owe you a pig in some manner and gives it to the person the debt is owed to. That can then be used to settle a debt they owe to somebody else.

We use debts we own to settle debts we owe based entirely on our natural ability to know who owes us a favour and who we owe a favour. Write them down and exchange them and suddenly you have a money system. Get one trusted guy to issue his iou in exchange for the one you have and you have a banker doing discounting and a standard unit of exchange in an area.