That's not really true. The runner can run all he wants. But he has to either be tagged, or the base where he was when the ball was hit has to be tagged, or he has to leave the field of play to be out.
For example, on a fly ball that is likely to be caught, it is common for a runner to advance half way to the next base (in case the ball is not caught), then if the ball is indeed caught, he returns back to the original base. He is not out.
No read the book. It has to do with what is considered proper advancement of the base runner. The runner may NOT start running before the ball is caught and the penalty is in fact that the runner is out.