The game is fun and simple: pretend you believe in the Flat Earth to piss people off. Refute their scientific facts with questioning and amusing statements, like "I never said it wasn't a circle" or "How do you really know Australia exists if you haven't been", etc. You win the game if the other person gets mad and truly believes you believe in the Flat Earth. You lose if either they don't believe that you believe this, i.e. if you've given it away with your facial expressions, or if they manage to actually present convincing evidence (e.g. like Carl Sagan's sundial shadows) that cannot be refuted.
I have conversed with both kinds. The latter can be highly amusing people to talk to, the former are legitimately strange and I can't fathom how they come to exist. The linked post may be correct; I think some people simply have the innate urge to reject everything they've been told and to look for a countercultural answer instead, regardless of the subject matter. At least this is a safe direction to channel their madness into.