EDIT: Apparently people don't understand that using gravity to generate energy is a thing, and the title "Powered Solely" suggested someone found a way to do that.
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.
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.
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.
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:
However,
Yes, Stirling Engines can be wonderfully efficient, and they are a delight to watch in action. Lovely machines.
You wouldn't do it for $5 though. Efficient Stirling Engines with a reasonable amount of torque actually require significant engineering, and that comes at a cost. However, they wouldn't require significant maintenance, they would provide power all the time that could be stored (again, at a cost), and might be able to trickle power an entire village.
Shouldn't it be said that it is powered by vacuum energy, since that is what kicked of the Big Bang? (And, before anybody tells me I'm wrong, let me say, "I'm wrong!" Instead of telling me I'm wrong, educate me. I love learning about this stuff, and I know my knowledge is weak.)