When I looked at the video and the chap held the LEDs closer to the camera it looked like LEDs had been modified and the plastic package was longer than it should be for a 5mm LED. Red was picked on purpose since it has the lowest forward volt drop, ~1.8V.
By cutting the LED close to the base of the plastic enclosure you could place a very small pick up coil, very small diode and very small capacitor inside the LED package and then glue it back together with the base & wire legs of another LED that was chopped in half, so it looks like an untampered LED (except the package is slightly longer).
If you look closely at the image of the plastic fascia when hes moving the camera around after the experiment, the fasica looks deformed as if there is something underneath the surface. This deformity could be a result of larger coils that have been created below each of the LEDs which in turn is connected to a circuit generating an RF field.
Due to Magnetic coupling the larger coil under the fascia causes a voltage to be induced in the coil inside the LED which is what is powering the LEDs (same as a Wireless Charging Mat for a Cell Phone)
The switches & the battery (& its clip) are probably a clever ruse to make you think they are carrying out the power and on/off control when in reality hes probably using another coil antenna under the fascia connected to an RF receiver IC (eg ~433MHz) to send on/off controls from someone else in the room.
You've been able can get very flat batteries for some time which could easily fit within the width of the plastic fascia which was shown to be about 1.5mm thick. Ultra thin Lithium Polymer batteries can be as little as 0.3mm thick so a cavity in the fascia could easily hide this battery.