It's not one of those trick that says "Kids, don't do this at home", at least we wouldn't put it in that category.
So, how is this done?
Starting with the fact that candles generate a lot more heat than light, it's not at all unreasonable to harvest some heat and turn it into light. The candle power increases the thermal energy to light conversion efficiency of a candle by placing a Peltier element over it. The heat causes the peltier element to generate electricity, which creates electricity.
A 1 kilogram candle contains 40 MJ of combustion energy, and might burn for 100 hours with a power of about 100 W. Thermo-electric conversion is pretty inefficient, but you should still be able to get a good part of a Watt to light the LEDs.
For greater effect try it with bigger candles and you get bigger electricity.