Here's why:
* It's the specific name of a game. Just like "Monopoly" or "Chess", it's a unique and specific name given to a particular game.
* Proper nouns are capitalized. This is a clue that it's a proper noun.
You could say, "I like to play the game of chutes and ladders," where "chutes and ladders" is acting as a noun phrase, but the actual name of the game is still a proper noun.