On this model the black wires attach to the bottom of the generator where the press on cap is. Stick the stripped ends of the two wires (headlight and tail light) in there and press the cap on. It should be a press and twist lock action. The bike itself is the electrical ground so there is only 1 wire that runs to the headlight and another to the tail light. They are wired in parallel, not series. There should be a small pointy screw in the generator mounting bracket that is the ground connection. I see it is missing on yours. It will dig a small hole through the paint to make electrical contact to the frame. it might be making contact anyhow. You can check that with a multi meter. Yes, the electricity flows through the headset bearings to get to the headlight. Looks like a white wire there. Perhaps that is a replacement wire going to the headlight? OEM were either black or grey.
These generators work by rubbing on the side wall of the tire. There is a mechanical lever that releases the catch. A spring holds it against tire. Lots of mechanical drag there. Not very efficient at all. Like it takes 100 watts of your leg power to generate 3 watts of electricity. It will knock a few mph off your riding speed. The 12 volt German versions are especially cruel. The bulbs are typically 6 volt 2.4 watt headlight and 6 volt 0.6 watt tail light. When one burns out the other gets full power and will also burn out. Riding over 15 to 18 mph will fry the bulbs. Some are rated for full power at 20kph (12 mph). There is NO voltage over load protection in the system.
Aligning the generator is important. It looks a bit out of line in your photo. The axis should be on a direct line to the wheel axle. The drive wheel should rub on the tire in the center of the drive wheel, not at the edge of the drive wheel otherwise it will destroy your tire pretty quick. They used to make rubbers for the drive wheel. Those would help protect the tire and also reduced the rotational speed (and power output) of the generator. Less light at slow speeds but it takes a higher speed to fry the bulbs. finding correct replacement bulbs can be difficult. Higher power bulbs just won't light up. Smaller wattage bulbs will burn out pretty much instantly.
My local bike coop has boxes of these very obsolete generators. Consider them for show, not actual use. Modern LED lights are vastly superior. There are certain brand/models that some Schwinn purists will pay big bucks for. Most have no value other than curiosity for younger cycling enthusiasts.
