Why would batch normalization allows us to use higher learning rate in the neural network?
Section 3.3 of the original batch normalization paper has a good explanation of why this is the case. 

Problem with a higher learning rate

First you need to understand the problem with higher learning rate. Higher learning rate causes exploding or vanishing gradients. In other words, gradients get multiplied by each other, so lower layers experience a compound effect of the gradients that are in higher levels. 

How does batch normalization help?

Batch normalization is all about keeping the activations of all layers normalized, preventing them from becoming too large or small. So this directly helps to prevent exploding/vanishing gradient. Due to this reason, batch normalization allows higher learning rates.