I would expect the unit to have three connections:
1. input to tail light
2. input to brake light
3. ground
Assuming your ground connection is good, then applying 12v to each of the other two should light up that LED or group of LEDs. You shouldn't need any other external components to make that work. A bad ground connection could make odd things happen, but I'm not sure that's your problem, or your only problem.
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