This LED cube has 512 LEDs. Obviously, having a dedicated IO port for each LED would be very impractical. You would need a micro controller with 512 IO ports, and run 512 wires through the cube.
Instead, LED cubes rely on an optical phenomenon called persistence of vision (POV).
If you flash a led really fast, the image will stay on your retina for a little while after the led turns off.
By flashing each layer of the cube one after another really really fast, it gives the illusion of a 3d image, when int fact you are looking at a series of 2d images stacked ontop oneanother. This is also called multiplexing.
With this setup, we only need 64 (for the anodes) + 8 (for each layer) IO ports to control the LED cube.
In the video, the process is slowed down enough for you to see it, then it runs faster and faster until the refresh rate is fast enough for the camera to catch the POV effect.