I see you had the same fun we all have with the wonky picture adding part of this site.
In looking on quadratech and reading the instructions for both apollo KC lights, you are intended to use the provided relay to connect them directly to the battery. Then based on the laws of your state tap in the appropriate wire for the switched lead on the relay.
If you are unaware, a relay is a switch. However, instead of you physically flipping the switch it is activated by feeding power to it. So if you wire the lights to the battery through the relay, then attach the switched lead to (for example) the fog light power wire, then anytime the fog lights are turned on, power will be sent to the relay, the circuit with close, and give power to the lights from the battery. They are made so a lower power circuit can turn on a higher power one.
I'm almost positive that there isn't enough power allowed through the fog light circuit for them to work. That is why the relay is provided. These lights draw too much power, the resistor is just making it worse. So the computer is killing the lights immediately to protect itself.
I'm not sure where you saw that they were plug and play, because quadratech puts these at a level 3. You have to do some wiring. But it can be done without putting a separate switch inside. This is for both the 55Watt 5" and the 100Watt 6 inch.