In the first relay position is in its native (NC) position, the electrolytic capacitor is discharged. The relay is not active.
When the 1st relay is flipped to the +12V position by the Anthem, the electrolytic capacitor is charged to +12V via the relay winding resistance. The pulse charge current through the capacitor and the relay will close the relay switch momentarily. When the relay is flipped back to ground (the Anthem turns off), the capacitor will discharge, again activating the 2nd relay momentarily. Thus the Anthem trigger can be transferred to the Proceed as if a key was pressed momentarily. A 330uF (micro-Farad) capacitor and 12V relay (260 Ohm resistance) will give about 0.2 seconds ON-time. In my case, a 1000uF capacitor provides about 3 times that time. The time is not really critical for the Proceed so long as it is greater than 100ms. I just wanted it to be long enough so I could see it on my digital multimeter to test it.
Anyway, that's the best explanation I can come up with. I originally thought that your approach was necessary, but an extensive web search suggested this, which does work. I was skeptical too until I built and tested it. Also also thought a resistor to bleed off the capacitor was necessary, but this kills too birds (triggers) with one stone.
If you want to try testing it, just build this:
I know that looks stupid simple, but it does work. Anyway, I'm not an EE so this is hurting my brain.