The power wire to the relay acts as a "signal". So stock configurarion has it hooked up to the low beam positive, which tells the relay to turn on the fogs only when the low beams are on (given the switch is on).
So with this mod, you are basically giving the relay a different signal source, such as your cigarette lighter. This turns on when you put your car to ACC, so if your fog light switch is on, they will turn on when u turn the key.
Other possible "signal" sources are: parking light - so the fogs turn on when u turn the light dial one click (but bad if you have an alarm that flashes the parking lights, unless you remember to turn the fog switch off before leaving your car). Or as the install says, the ignition, so they turn on when you start the car.
The reason for the relay having 2 red/white wires, I am guessing is because one is the low beam, the other is the high beam. If you havent noticed, when your fogs are on (and low beams on) and you hit the high beam, the fogs turn off to comply with DOT laws (only can have 2 pairs of lights on at any time). I think the high beam wire that goes into the relay tells the fogs to turn off. Im not sure though, im just guessing. So doing this mod would eliminate that?