Holy cow, I haven't seen a lighting setup like that for at least 30 years. :-) Without a wiring diagram, nobody's going to be able to tell you what it's supposed to be doing.

However, I would guess that the bottle-gen is intended to charge the batteries, and they'd be used as the power source light a stand-light. If those are lead-acid batteries ( which I suspect they are ), then the generator isn't likely be generating nearly enough voltage to charge them directly. It looks like the generator claims "12V" on the side, but you need approximately 14.1V to charge the battery. If the batteries are more than a couple of years old ( or much less, if they're being improperly charged ) then they're probably shot, anyway.

This is the sort of setup where you can immediately evaluate the quality of the work by assuming an inverse relationship between the amount of cheap electrical tape and construction quality. ;-)