After some tinkering, I fixed it.
I went into the game directory and clicked on "SOTE.EXE".
After a short wait, the old style Lucas Arts game launcher came up and I clicked on "Play".
That opened the game configuration screen and I created the player, set the degree of difficulty and chose the screen resolution and just minutes later I was downing an AT-AT with my tow cable :)
You shouldn't need to have to do that, though. The GOG launcher (on first run of the game) opens up a dialog where you set the difficulty and the resolution (and some other options like fog and such). The Play button is indeed greyed out, but right next to it is a New Player button which you click on and choose a save game location (which I put inside the game installation folder). After doing that the Play button is no longer greyed out. Sounds like you just did that part inside the game itself rather than the launcher.