As an aside to argonaut's response (which is correct) it is important to note that there is a lot of ongoing work in understanding what the hidden layers in a CNN are doing. For image-classification type problems, it's pretty easy to pass an image in to the network and output the features each layer extracts.

Indeed, many recent systems rely on CNN-extracted features being passed to a SVN classifier to classify images that the CNN wasn't trained on (because the SVN is easier to train will less data, and using the CNN frees one from doing feature engineering).