It's the title of a book, as well as the name of the popular television adaption of said book. Within the context of the book/ show, "a game of thrones" is what the politically connected individuals call their various plots and machinations - they refer to the sometimes deadly power struggles of court as "a game", partly in jest, partly because they're sociopaths.
The cardboard castle here is a rough recreation of the King's capitol fortress, as depicted on the TV show. The chair made of carrots is a reference to the Iron Throne, an iconic setpiece from the show that's frequently seen in promotional material. It is the King's throne, notable for being made entirely out of sharp, pointy swords.
Oh, I was just joshin'. My room mates are obsessed with GOT so I have to constantly hear about it. I give zero fucks about it, but I've been forced to osmotically absorb the whole shitshow. The only part I've ever actually seen was the Red Wedding scene. That part was pretty funny, I guess.
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