The key to that system, it seems, is its membrane electrode assembly (or MEA), which contains a material that's capable of breaking down water into hydrogen and oxygen through a chemical reaction. Not surprisingly, the company isn't getting much more specific than that, with it only saying that it's adopted a "well-known process to produce hydrogen from water to the MEA." Currently, that system costs on the order of ¥2,000,000 (or about $18,700 -- not including the car), but company says that if it can get it into mass production that could be cut to ¥500,000 or less (or just under $5,000)
There is a video from Reuters that I have tried to embed below, but I am not sure if it will show up on the final blog (here's the link to the Reuter's page that houses the video):
Update: Looks like there's more to it than meets the eye. See this discussion on Slashdot.