Step 7: Circuit time!
The idea is the 555 timer generates a square wave which charges up the capacitors in the on phase. In the off phase, the capacitors light the LEDs, depleting over time and creating the fade out effect. Theoretically, during the on phase, the current used to charge the capacitors should detract from the LEDs, creating a fade in effect, but often the current is so large that this is nearly instantaneous. Alternatively, you can put in a switch that bypasses the pulse circuit (so that the ground from the LEDs goes straight to the mains ground) so you can go from pulse to steady on.
I more or less used capacitors and resistors that were lying around. You can customize the frequency using the 555 timer calculator. Alternatively, you could sub in a potentiometer for RB to get a variable rate.
The 12000 uF capacitor is just a bunch of large 12V capacitors wired together. You may need more or less depending on the number and rating of the LED strings to get the right effect.
Test out the circuit on a breadboard connected to the car's electronics before putting it onto a PCB. And then test it when you've soldered it before boxing it up.