Why would anybody want to use "hidden layers"? How do they enhance the learning ability of the network in comparison to the network which doesn't have them (linear models)?
"Hidden" layers really aren't all that special... a hidden layer is really no more than any layer that isn't input or output. So even a very simple 3 layer NN has 1 hidden layer. So I think the question isn't really "How do hidden layers help?" as much as "Why are deeper networks better?".
And the answer to that latter question is an area of active research. Even top experts like Geoffrey Hinton and Andrew Ng will freely admit that we don't really understand why deep neural networks work. That is, we don't understand them in complete detail anyway.
That said, the theory, as I understand it goes something like this... successive layers of the network learn successively more sophisticated features, which build on the features from preceding layers. So, for example, an NN used for facial recognition might work like this: the first layer detects edges and nothing else. The next layer up recognizes geometric shapes (boxes, circles, etc.). The next layer up recognizes primitive features of a face, like eyes, noses, jaw, etc. The next layer up then recognizes composites based on combinations of "eye" features, "nose" features, and so on.
So, in theory, deeper networks (more hidden layers) are better in that they develop a more granular/detailed representation of a "thing" being recognized.
Hidden layers by themselves aren't useful. If you had hidden layers that were linear, the end result would still be a linear function of the inputs, and so you could collapse an arbitrary number of linear layers down to a single layer.
This is why we use nonlinear activation functions, like RELU. This allows us to add a level of nonlinear complexity with each hidden layer, and with arbitrarily many hidden layers we can construct arbitrarily complicated nonlinear functions.
Because we can (at least in theory) capture any degree of complexity, we think of neural networks as "universal learners," in that a large enough network could mimic any function.