Using an arduino-compatible hardware platform that is stackable, this particular cabinet use an open-source circuit called a TinyArduino. Created by the folks at TinyCircuits, these bundles of circuits are about the size of a quarter. The company makes a number of kits, including an arcade cabinet version with a built-in LED. As of right now, you can get the files for Space Invaders and Flappy Bird off of their website here.
Want to try your hand at building one? You can get the kit off of their website here.