A lot of the benefit in deep neural networks comes from the ability of lower layers to learn representations that the higher layers can then use to perform their classification.
In a typical image recognition setting, the first layer might learn simple components of an image, like edges, the next layer might learn more complicated features that are combinations of edges, like shapes etc. and the 3rd layer can learn combinations of shapes, for example it could learn face like features if performing face detection, or it might learn components of vehicles (like wheels, headlights etc.) in a vehicle recognition task.
This image demonstrates this type of learning:
That is of course the ideal case. Feature learning and representation is an active research field and learning these types of features requires a lot of effort and domain knowledge.