Turn: "Save Game" button...

[Dreadfeather]

... what's it for?
Seems like my moves are auto-saved the moment they're made.

[SnotlinG]

It will verify all your moves have been saved by the servers.
Sometimes (very rarely) if you have a laggy connection and move units without exploring FOW a move could be lost.
If you hit save game button it will verify/resend the move information to make sure it has the latest status