Would someone be able to explain to me or point me to some resources of why (or situations where) more than one hidden layer would be necessary or useful in a neural network?
Basically more layers allow more functions to be represented. The standard book for AI courses, "Artificial Intelligence, A Modern Approach" by Russell and Norvig, goes into some detail of why multiple layers matter in Chapter 20.
One important point is that with a sufficiently large single hidden layer, you can represent every continuous function, but you will need at least 2 layers to be able to represent every discontinuous function.
In practice, though, a single layer is enough at least 99% of the time.
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