Based on nothing but what is on the linked page. . .
To move energy across an air gap you need an alternating current. The power supply probably is a wall-wart, with an oscillator and power amplifier are in the base unit, and it has a clearly marked on/off switch location. This could be a membrane switch or a capacitive touch switch circuit, but either way that covers the power and control parts.
The light can be turned off but still levitate, so there might be a very low power wireless data link between the base and the bulb. With a $1 microcontroller, a ton of things are possible.
There is a wire coil in the light bulb base that acts as the secondary of an air-core transformer. Depending on the LED string construction, either the AC waveform is applied directly to the LED strings (with strings mounted in opposite-conducting directions), or rectified and applied to the strings with all strings conducting in the same direction. Either way, the strings are flickering too fast for the eye to perceive.
In the video, lights are rotating in both directions. This suggests that there is nothing guiding the bulb's motion.
The text implies that the levitation and power transmission are done with separate systems. Here is what I hope they are doing:
The drawing shows four electromagnets in the base. There also could be neodymium magnets in the base and bulb to help with levitation, but the main levitation probably is produced by an aluminum disc in the light bulb base. The AC field induces eddy currents in the disc, and those currents produce their own field that opposes the base's field. This is captured by a minus sign in Faraday's Law of Induction.
Something like this:
https://www.youtube.com/watch?v=AHED5xSnnM8
https://www.youtube.com/watch?v=A7BScXvM8w0
OTOH, the levitation could be done 100% with permanent magnets, and the electromagnets are used for power transmission only. If so, this makes power control very simple; turn up the output of the power amplifier, bulb gets brighter. Turn the power amplifier down or off, and the brightness goes down or off. No brains needed in the bulb base.