Stirlings are a member of the class called "hot air engines". As a group, hot air engines work by exploiting the expansion and compression of a gas when it is alternately heated and cooled. There's a good explanation/animation here.
(That particular engine is a "true" Stirling because of those green cross-hatches. That's a "regenerator" that vastly improves efficiency. However, most people call all generic hot air engines, regenerator or not, "Stirlings".)
Stirling engines are awesome for a lot of reasons, but one of the best reasons is that you can use any source of heat. This isn't an internal combustion engine that can use only one type of fuel that's later impossible to wean away from. This is an external combustion engine. You could use natural gas or wood or ethanol or solar energy or geothermal energy. In fact, some satellites/probes use Stirlings to convert nuclear energy to electricity. That's right, Stirling engines in spaaaaace.
Anyway, back to Earth and me. Because Stirlings are so simple and efficient, you can make even a really crappy one run fairly well. They are very common first projects in machine tool classes, for instance, whereas building an IC engine from scratch would be...challenging for a beginner. However, I'm not even at the level of machine tool student, so even that option isn't open to me.
Fortunately, lots of people have been coming up with plans for engines that don't require machining. A few years ago, I tried one that used water-based pistons but I couldn't get it to run. I really have no idea what the problem was, probably multiple things. More recently, I found some "plans" for a test tube Stirling. It took me a while to actually get it going, because it turns out that glass syringes are incredibly finely made and also very rare nowadays. I actually had to buy one on ebay as an antique!
I think the functioning is clear enough to explain itself. (Note: This is technically not a Stirling since I have no regenerator, although I could stuff a bit of steel wool in there to fix that.)
After viewing the video again, actually that might not be so clear. Here's what is happening:
- Candle heats air inside left end of test tube.
- Air expands, pushing "piston" up.
- Marbles roll to the left, displacing the air to the right.
- Air cools down, contracting.
- Pulling piston down and causing marbles to roll right.
- Air displaces leftward.
- GOTO 1