I believe, this post and this post written by me well address your question.
In fact, this is a very interesting question that you ask. It deserves some more explanation.
1. Fully connected networks
The more layers you add, the more "nonlinear" your network becomes. For instance, in the case of two spirals problem, which requires a "highly nonlinear separation", the first known architecture to solve the problem was pretty advanced for that time: it had 3 hidden layers and also it had skip-connections (very early ResNet in 1988). Back then, computing was a way less powerful and training methods with momentum were not known. Nevertheless, thanks to the multilayer architecture, the problem was solved. Here, however, I was able to train a single-hidden layer network to solve the spirals problem using Adam.
2. Convolutional nets (CNNs)
An interesting partial case of neural networks are CNNs. They restrict the architecture of the first layers, known as convolutional layers, so that there is a much smaller number of trainable parameters due to the weights sharing. What we have learned from computer vision, moving towards the end of CNNs layers, their receptive fields become larger. That means that the subsequent CNN layers "see" more than their predecessors. Conceptually, first CNN layers can recognize simpler features such as edges and textures, whereas final CNN layers contain information about more abstract objects such as trees or faces.
3. Recurrent nets (RNNs)
RNNs are networks with layers which receive some of their outputs as inputs. Technically, a single recurrent layer is equivalent to an infinite (or at least large) number of ordinary layers. Thanks to that recurrence, RNNs retain an internal state (memory). Therefore, it is much more difficult to answer your question in the case of recurrent nets. What is known, due to their memory, RNNs are more like programs, and thus are in principle more complex than other neural networks. Please let me know if you find an answer to your question in the last case.
To conclude, the higher number of hidden layers may help to structure a neural network. Thanks to the recent developments such as ResNets and backpropagation through time, it is possible to train neural networks with a large number of hidden layers.