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In the 1980s many movies explored the fears of nuclear war with the Soviet Union. The quote "Nuclear war: strange game, the only way to win is not to play" comes from the movie War Games (1983) starring Mathew Broderick. In this movie a teenage boy hacks a military computer looking for a video game and almost starts WWIII. This line comes from the computer, when it finally understands there is no way to win the "game" nuclear war. The quote means that no matter who wins a nuclear war everyone will lose because the level of destruction is so huge.
Sources: Housman, D. (2006, March 6). Attack of the killer Exersaucers [weblog post]. Dan Hausman Hypercritic.
“A strange game. The only winning move is not to play” [weblog post] My Geek Wisdom: Because There is A Lot of Knowledge to Get Out of the Weirdest Things.
"War Games (1983). IMDB.