A flame-licker, or vacuum, engine is a variation of a Sterling engine. It works by opening a valve during the down stroke of a piston which allows heated air to enter the cylinder. The valve then closes and the air rapidly cools. This causes the air to contract which draws the piston forward for a portion of the stroke which is all that provides power to the engine.
When the pressure inside and outside of the cylinder equalizes, a valve opens which converts what was the power stroke into the exhaust stroke and the remaining air is vented. This piston is aided by a flywheel which keeps it moving during all but the very brief period of the power stroke.
Needless to say, it is a very inefficient engine and can't power much more than a toy.
The only detailed description of any sort I could find of the flame-licker engine was the article Mystery Engine Breathes Flame (page 206, of the January, 1950, issue of Popular Science). While the bulk of the article discusses how to machine and build a flame-licker engine, if you're interested you can dig through it and get a clearer idea as to how it works than my brief description provides.