What does it mean to feel stuffed to the gills?

If you go fishing or even just go to the fish market, look down into a fish's mouth. You will see the gills on both sides and just after the gills in the oral cavity you will see the opening of the gullet. If you have a particularly well-fed fish, the gullet may show unswallowed food because the stomach is too full to hold more so, it is waiting to be swallowed. Humans don't have gills but the analogy can be appropriate when you feel you have swallowed more than your stomach can hold.