[moridinamael]

Think of each layer as an opportunity to perform a level of abstraction or categorization. Concepts are built out of smaller concepts which are built out of smaller concepts; lots of layers allows lots of hierarchy in the concepts.

The first layer might recognize and respond to pixels in particular parts of an image, the next layer will group certain of those pixels-responses together into an abstraction you might call a "line", the next layer will respond to certain groupings of lines and add a level of wiggle-room regarding where the lines are in the image, and the final layer will judge whether a combination of groupings constitutes the letter "A". Or at least, if you spent a bit of time poking at a deep network, giving it slightly different inputs, you might eventually conclude that this is what the layers were doing.

Without layers, you're basically just approximating a simple function or mapping with one level of abstraction.