I couldn’t find the original paper which proposes the importance of weight normalization.
However, the reason is simple. The network can be converged with the values in range of [-1, 1]
Here’s a sequence of multiplication of 3 numbers

10 x 10 x 10
0.1 x 0.1 x 0.1

Convergence/divergence is important. We can easily handle the value of 0 but not 1000000000000…