Technically the term is cloacal respiration, and it’s not so much breathing as just diffusing oxygen in and carbon dioxide out, but the fact remains: when turtles hibernate, their main source of oxygen is through their butt.
As cold-blooded animals, when the temperature drops in the winter, a turtle’s internal temperature drops with it, and its metabolism slows down to match. While they are in this slowed-metabolism hibernation period, their oxygen needs are quite low, and the oxygen diffused from the water running over them is enough to sustain them until spring. If times get really tough, they can always switch to anaerobic respiration: powering their metabolism without oxygen, but this mode comes with a time limit due to the buildup of a respiratory byproduct, lactic acid.
This breathing process is fairly common amongst amphibians and reptiles and is properly called cutaneous respiration. Besides the turtle butt-breathers, notable users of cutaneous respiration include frogs, salamanders and sea snakes.