Does she put it in the fridge before using it or something? Or does it use the difference in temperature between your hand and the flashlight.
If you RTFA you'll see she's using the aluminum flashlight body as a heat conductor and the "head" and other exposed portions of it as an air-cooled heatsink.
She's stuck the handle of the light into an insulating plastic pipe, cut a hole in the pipe, and stuck the peltier cell in the hole, with the "cold" side in contact with the flashlight handle and the "warm" side in contact with the hand. (I expect the next step is to wrap an outer aluminum tube around it to conduct heat from the whole hand to the cell, rather than just heating it with a patch of palm directly contacting it.)
Voltage boost converter between the peltier assembly and the LED (because the peltier cell she used was not stcked for the right voltage to drive the LED.) The LED shines as long as you hold it, if the air is cool enough. (She's used it for 20 minutes running.)
Also, since this is generating electricity from a temperature differential, rather than generating a temperature differential from electricity, wouldn't this be the Seebeck effect?
Yes. Seebeck discovered current generation from heat differential (with dissimilar metal wires and a compass needle), then Peltier discovered heat-pumping with current.
But, like most rotating electric machinery (where the same device is a motor or generator depending on whether you power it or twist it), the same effect is a heat pump or heat engine (depengding on whether you apply a temperature difference and pull power or apply power and pump heat).
The effect is now often called the "Peltier-Seebeck effect" in textbooks. The cells are typically called Peltier Cells because the efficient ones are manufactured mainly for heat-pumping, though they work just fine both ways.