The machine converts gravitational potential energy to electromagnetic radiation. People generate the gravitational potential energy by lifting the weight using kinetic energy, which was likewise created by converting chemical energy.
It's easier just to say "Gravity" because, without gravity, it wouldn't work with its inanimate weight.
Actually, it does, and that's the perfect way to analyze this. Food is good for keeping humans alive, but it's a pretty foolish fuel to burn in a human engine to accomplish things. It's perishable, it's eaten by rodents, it's hard to transport, and more. I chuckle darkly when someone tells me that something is "green" because it's human powered. Growing food isn't free - it uses conventional energy sources to harvest, transport, etc., and it's a ton less efficient than just burning that fuel directly.
GRAVITY!!!
The ingredients needed for fusion on the other hand are thought to have been created during the some of the early stages of the Big bang so fusion is truly independent of stars.
At any rate, the phrase was written by Enpundit, not GravityLight. Their indiegogo page is less dramatic and more scientific about what powers their lamp: