What you see above is a generator that converts heat to electricity. [Reukpower's] thermoelectric lamp is one of those hacks that makes you scratch your head even though you understand why it should work. The heart of the system uses a Peltier cool, just like the thermoelectric solar generator. When there is a temperature differential from one side of the Peltier to the other a small current is generated.
In this case a candle heats one side and a heat sink cools the other. The tiny voltage picked up from the Peltier’s contacts is then boosted using a joule thief. We’ve seen LEDs powered with a joule thief before, benefiting from their own low power consumption. In this case, the boost circuit is scavenged from an emergency phone charger and probably achieves higher efficiency than if he had built it himself.