OK, those pictures help quite a bit. The wiring diagram will be very helpful, I'll explain it to you. There are only 3 wires we need to hook up.
White wire with ring on the end gets connected to your positive terminal on the battery.
The black wire with ring on end connects the the negative terminal on your battery.
So, for the lights to work off your stock fog light switch, you want to unclip the plug that goes to the on/off button in your last pic. Then using the little red plastic wire adapters that came in your kit, you will need to connect a single wire (purchased separately) to the white wire that goes into the relay from your switch plug (not
the white wire that is cut at the end). Then, you will need to use another wire connector to clamp onto your stock fog light wire. I am still trying to find out what color the fog light wire is as there are 2 wires at each fog light.
Honestly, you could try one, and if it doesn't work, then try the other, but that's up to you. It won't hurt anything if you get the wrong on the first time.
Here is a page showing how to use those little wire connectors (I found one with pictures for ya
I have no idea what that site is about, but scroll down till you see the little series of how to use the blue wire tap with some pliers. You can pick some of those up at Lowes or Home Depot, as well as some wire to run from your stock fog light wire to the relay wire. The size is measured in AWG, and I would suggest finding some 18AWG on a spool that they can cut for you. 6 feet should be more than plenty.
Oh yea, don't forget to plug in the lights once you're done