Can you explain more precisely what point you're confused on?
I'll provide an example for corn, since I'm somewhat familar with it. When you buy a futures contract, you can hold it with the intent of buying the underlying commodity. Think of this as buying from the Exchange, as at expiration you would drive up to an Exchange delivery point (picture a grain elevator) and in rough theory, you would hand the elevator your long futures contract (purchase of underlying corn) and they would load out your truck with corn, you would pay the exchange the price you bought futures at and go about your merry way.
When you bought the futures contract, an additional party sold that contract to you, I'll call him party B. Party B sold contract of corn to you, THROUGH THE EXCHANGE. When the contract expires, party B must drive up to that same Exchange delivery point and deliver his corn at the price he locked in his futures sale at.
See how it works? For normal delivery, you go through the exchange. If you do exchange for physicals method, the buyer and seller of a futures contract (bought and sold thru the Exchange) exchange corn on their own terms, away from the Exchange delivery point. The Exchange must approve of this though, because the original contract was done through the Exchange and expects them to deliver / pickup at their delivery point.
Hope this helps.