If I'm understanding it correctly this generator has like wires out that then connect to the lights or something right?

If that's the case you'll need to see what kind of voltage it's putting out to those wires and then you would want to splice those wires into diodes to prevent the current from leaking backwards out of the capacitors. Then attach as many capacitors as you need for the voltage that it is providing, remembering that it is okay to have higher voltage ratings but not lower voltage ratings.

Chances are this will cause the light to be less bright as the capacitors are charging, so for the first few moments when you are riding it probably will be less bright but after the capacitors reach a steady charge you should see the light take a much longer time to dim down.