Something billows when there's a fluid or blowing motion, such as the air filling a curtain at an open window, or smoke billowing from a fire.
This word originally meant a wave, and that image should help you remember its current meaning too. The word usually suggests movement and growth, such as waves building and crashing. There are certain clouds that look like ocean waves, and so are called billow clouds. If you have a big shirt, the wind might make it fill with air and billow. Balloons are billowed when you inflate them. Billowing can also mean to move with difficulty — slowly.