I have been invited to a private game.  Once I've chosen my colour I have clicked "Join Game", but the next screen say "You have successfully left the game" and at the bottom it says "Sorry, you have not been invited to this game".

Any ideas how to fix it?

I confirm other people have managed to join the game so it is not an issue with the host's setup.

i know that i've been in your situation, but it was because i hit the Decline Game button. You may have done it accidently. Its right next to the drop down button for choosing your color, so its easy to do without realizing it.

I believe the host can re-invite you to the game. Might give that a shot.