First of all, hidden layer in artificial neural networks a layer of neurons, whose output is connected to the inputs of other neurons and therefore is not visible as a network output.

Now, let me explain the role of the hidden layers on the following example: There is a well-known problem of facial recognition, where computer learns to detect human faces. Human face is a complex object, it must have eyes, a nose, a mouth, and to be in a round shape, for computer it means that there are a lot of pixels of different colors that are comprised in different shapes. And in order to decide whether there is a human face on a picture, computer has to detect all those objects.

I think the following diagram worth a thousand words.

Basically, the first hidden layer detects pixels of light and dark, they are not very useful for face recognition, but they are extremely useful to identify edges and simple shapes on the second hidden layer. The third hidden layer knows how to comprise more complex objects from edges and simple shapes. Finally, at the end, the output layer will be able to recognize a human face with some confidence.

Basically, each layer in the neural network gets you farther from the input which is raw pixels and closer to your goal to recognize a human face.