My guess is that the mechanism used a Geneva wheel to advance the film one (or some other odd number) of frames. When the film was in place, a small light bulb would probably have lit for a few moments, allowing the viewer to read their fortune.
Here's a tantalizing view of the actual mechanism:
You can see the old battery in the lower right. Hey...are those thread spools on which the film sat? I would love to get a chance to fix this piece up.
Here's the listing on eBay for the fortune teller automaton with answer strip and coin mechanism.