When the author mentions "normally distributed numbers" it sounds like he is sampling a normal (Gaussian) distribution. In fact, he is sampling from a uniform distribution.
The gaussian distribution is invariant under rotations in 3d (and more generally invariant under any orthogonal transformation in any dimension), so if one takes a random vector X whose components are independent N(0,1) random variables and compute X/|X| (where |X| is the usual euclidean norm of X) the result is guaranteed to be uniform on the unit sphere.