What is meant by following-through in bowling?

Answer 1

Assuming you have a typical pendulum type swing, the ball is release just as you are coming upwards in the swing. If you stop your arm movement right after the point of release, it can sometimes interfere with accuracy, distance, roll, etc. By continuing the pendulum swing post release, you improve accuracy, distance, roll, speed, etc.

Golf is another similar thing, where you see pro's continue the club swing after hitting the ball until the club comes completely behind them.

Answer 2

As you release the ball from the hand, the arm swing should continue instead of immediately stopping. This provides better accuracy.