I believe that this is done using Restricted Boltzman Machines[1] trained with the stylised image.
Think of it as a network that receives an image on the input layer, sends it to one or more hidden layers with less nodes (like an auto-encoder), and then tries to reconstruct the image on the output nodes. This is like a lossy compressor-decompressor overfitted to the stylized image.
Now, just pass the real image as an input to your network and the output should be a stylized version of the input.
I think Alex J. Champandard's implementation is probably the best one out there right now. It has a ton of knobs to twist and is very fast.
Aside from the 'missing image' part, what is the fitness example here for training? How does the training process determine what a good image is given there aren't many (any?) examples of a source image -> picasso mapping?