Human burps are air that you swallowed, or from CO2 from either drinks or produced when stomach acid mixes with bicarbonate. That CO2 does not necessarily has to come out as a burp, it can also pass into the blood stream. We don't have a rumen like cows do, where bacteria start breaking down the food. Our stomachs are mostly free of bacteria (except for things like helicobacter pylori). All our bacteria-activity is in the large intestine. A lot of the gases are absorbed and exhaled through the lungs, not all of it has to come out through the rectum.

Much better explanation and a lot more details: https://www.ncbi.nlm.nih.gov/books/NBK417/