Why Chess?
I suspect it has something to do with the ancient art of war. You see, in this day and age, warfare has become as impersonal as, well, the internet! "Push button warfare," where enemies can be dispatched at a tremendous distance via computerized weaponry, has nearly eliminated any personal skill in combat. The days of two heroes taking the field with one besting the other by sheer ability (what the ancient Greeks called monomachia) are over....
Except on the Chessboard, that is.
For Chess, being a creation of antiquity, manages to capture the essence of ancient warfare. Two players, serving as generals of their respective miniature armies, clash in a mental contest of strategic wits and tactical guile. Even though it is a bloodless contest, the violence is very real. To quote Paolo Maurensig:
"What occurs on (the chessboard), in the form of a creative act sometimes resembling a true work of art, is in reality a struggle of exceptional violence, a form of bloodless homicide whose outcome is shared by the contenders alone. Nothing binds two people like a serious challenge on chessboard, making them counterposed poles of a jointly produced mental creation in which one is annihilated to the other's advantage. There is no harsher or more implacable defeat. The players bear lifelong scars, neither body nor soul ever recovering fully. Anything that might reawaken memory of the mutilation is violently repulsed."
Ask anyone who has entered the martial arena of Chess and he will quickly confirm such a notion.
It is the pure struggle of will, this latter-day monomachia, that has proven to be so engaging over countless ages, from its birthplace along the Persian/Indian border, to its adopted homes of Western Europe and, eventually, America. You don't play a game of chess, you survive it.
So why Chess? It's a trick question. It's not a matter of why play Chess, but a reality of how can one not play Chess.

Fellow gladiators of the Chessboard, I salute you!