The design shown is again nothing new. We had one in the 1960s in the ME Thermo lab where I attended college. That one was a diesel cycle but that really doesn't matter, gas or diesel they are all running as two stroke engines. They are sometimes called "bounce piston" engines, named for the chambers at the extreme end of the piston. The sealed chamber allows air to compress and limit the stroke of the piston, stopping it and returning it in the down-stroke direction rather like a spring. It also how the engine is started, you inject compressed air into the bounce chamber to get the piston moving. After it fires on the down-stroke the cycle continues naturally.
As to how to take off power, there are several methods. The center piston shown in the animation at the website illustrates a double acting pump or compressor which could move either liquid or gasses. An alternative would be to reciprocate a magnet through the center of a solenoid coil, thus providing and alternating current output. Another method would be to use the exhaust gas as a heat source for a central heating plant.
And I saved the best for last, a piston powered rocket (reaction) motor. Way back when I was a pup, I was a regular reader of Science & Mechanics. One particularly memorable issue had a cover feature of a backyard built go cart powered by something called the "screaming demon". It was essentially a bounce piston engine with both the exhaust ports plumbed thru a convergent/divergent DeLaval nozzle. The nozzle was jacketed and a stream of water was passed thru it to keep the thing from melting. The water flashed to steam and was ported out of the throat of the nozzle to augment the thrust of the engine by adding mass to the hot gas flow. The thing was fueled with and acetylene/air mixture. If you remember your chemistry, acetylene does not take well to pressure. Anything much above 15psi or thereabout and it detonates. That normally is annoying but in this case it means you don't need an ignition system, you just valve a bit of compressed air to the bounce chamber and your off to the races. This thing could easily push the cart to 80 mph (like all rockets , the thrust continues to accelerate you 'till you chicken out). If you hit a bump, you can go "airborne". The only way to measure the speed of the engine cycle was to record the impressive exhaust note and try to beat the signal against a signal generator. Estimates ran over 100,000 cycles per minute! Anyway, that how you do it.
Regards,
GtG