At the beginning of a chain/swap (with the exception of the monthly egg swaps) the person starting the offers an item for swap. This is a donation to get the swap/chain going. After that:
Pretend that Ann has Apples for the current offer (after she accepted the previous swap). Bob wants the apples so he says "Mine". Bob offers bananas. He then sends a PM to Ann stating he claimed her apples and gives her his address. Ann ships the apples to Bob with a delivery confirmation number. Ann pays to ship the apples.
In the mean time, Connie is craving bananas so she posts "mine" and offers chocolate. She sends her info to Bob via PM and Bob ships the bananas to Connie with a delivery confirmation number. This chain just continues.
I will post more when I have time.