For the two devices to speak, the OUT or Talk, or Transmit or TX (or whatever you want to call it) on one device gets connected to the IN or Listen or Receive (or whatever you want to call it) on the other. And you need a ground too. So to have single direction communication, like from the GPS to the radio, you need two wires. To have bi-directional communication (GPS send to radio and radio sends to GPS) you need 3 wires. Some devices have separate ground for each port. While this is a good idea is it not necessary.
According to your color wiring diagram it should be
GPS Transmit and Radio Receive:
Garmin Blue (NMEA OUT / TX / Transmit) to Ray Grey (NMEA IN / RX / Receive)
Radio Transmit and GPS Receive:
Garmin Brown (NMEA IN / RX / Receive) to Ray Blue (NMEA OUT / TX / Transmit / Talk)
And the grounds:
Garmin Black to Ray purple (NMEA IN Ground) AND Ray Brown (NMEA OUT Ground)
Barry, thanks very much for your help.
You're welcome. I hope this helps.
Note that the important connection for DSC GPS and VHF is the GPS transmit and VHF receive. This is so the VHF radio can obtain your current location for emergency messages. Communication in the other direction (from VHF to GPS) is not as important and not all GPS units support it. That is used in case you want to see the location of another vessel on your GPS screen. Your radio can request the position of a friend, get that location in a coded message, send it to your GPS and your GPS can display the location. Is that useful? I don't know. Maybe if you're a fisherman and your friend is getting bites and you are not?
One last comment, NMEA 2000, a real network, with multiple talkers and listeners is making all the difficulty and slow communication of NMEA 0183 go away. This is happening now but will take a few years to be complete.