I bought the deluxe edition of Planetfall on this platform, and when I try and use Multiplayer it tells me I need to activate my copy in order to do so.

Where do I find my activation key for the GoG version, and how do I use it?

Go to your games list (on the web, not Galaxy).
You should see a tab called "serial keys" when you click on planetfall.
You will need a paradox account as well, I think...