When you fly fish, you cast forwards and backwards in a back-and-forth motion. While you're moving your rod forwards and backwards you're extending the fly line so it gets longer until you've reached the desired length. Then you let the line fall on the water in your last forward cast.
A back cast is the portion of the motion when you move your rod backwards. Given that you cast when moving your rod forward and not backwards, the back cast is also sometimes referred to as a 'false cast'.