The heat from the candle flash-boils the water that's in the "boiler". This is ejected out of the tubing, and once the diaphragm relaxes, it creates a vacuum which pulls more water into the boiler to be boiled again.

The reason it doesn't just jerk back and forth in this exchange is because it sends pressure out in one direction, but draws water in from all directions.

The wikipedia article has a subsection that is well explained IMO:

http://en.wikipedia.org/wiki/Pop_pop_boat#Principle_of_opera...