The system works by timing the slot in the platter. The Arduino uses an internal timer to clock each revolution. It achieves this using an infrared gate, which triggers a hardware interrupt on every full revolution of the platter. The Arduino uses the revolution time and phase to schedule a second internal timer. This second timer uses an interrupt to schedule the timing of the LEDs, firing tens of thousands times a second to build a stable, visible image.
This work is based on Ian Smith's version of the same. His work is excellent, and his website on this topic will help provide a complete picture for this instructable.