Oxygen dissolves easier in cold water. The fish will also have a slower metabolism in cold water, thus needing less oxygen. This does not mean they don't need ANY oxygen in the water, it just means most people don't have to worry about it as much during the Winter months.
My pond has a couple small waterfalls which I rely on to provide the oxygen content in the water. If I were to turn off my pump in the Winter, there would be no water movement, and thus very little chance of oxygen exchange at the surface of the pond. Obviously everyone's pond is different, so you have to consider your particular setup... If you turn off your pumps during the Winter, you still need something to keep the water moving some. An air pump is an easy solution because it does double-duty -- creating some movement of the water, and introducing fresh oxygen.
If you have an air pump running already, its probably providing what the fish need. An air-stone will increase the water movement, and thus the oxygen content (and it might even be more effective at keeping the open hole in the ice?) so its just one less thing to stress the fish.
