The idea is that their electrolysis unit is fed by the generator, which is in turn fed by the hydrogen from the electrolysis unit. To make the contraption somewhat more scientific-looking (and less suspect), they've put two water filters in the loop, presumably to 'purify' the hydrogen.

Does the article say that the generator runs the electrolysis? I don't see where it does.

A reason, though perhaps not theirs, storage. If you have excess wind/solar you could use a hydrogen generator to create enough to provide some minimal power for low production times or overnight.