It's not complicated. Let's say you're using a long piece of webbing to build the anchor. Before you tie the ends of the webbing to make a loop, you put the webbing through the rap ring. Then when you form the loop, the ring is linked into it. When you form the master point of the anchor (i.e., the loop that you would normally put a locking biner through), you do it so that the rap ring is there in the master point. Now you rappel off of the rap ring.
To make sure you don't lose the rope, you go through the usual procedure of tying the middle of the rope to something before you throw the ends down. Then before you rap off, you remove the knot so that you'll be able to pull the rope down at the bottom.
All you're leaving behind is the webbing and the rap ring, both of which are cheap to replace. Having the rap ring lets you avoid leaving behind a locking biner, which is more expensive. Sometimes you will see anchors that have been permanently left in place (e.g., on canyoneering routes), and what's left behind is a rap ring.