It comes from Rugby, where a player has to literally touch the ball down in the end zone for the score to count.
American football -- which evolved from rugby -- originally required players to touch the ball down, too. The rule was eventually eliminated, but the name of the score stuck.
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It originally came from football's ancestor, rugby. In order to score in rugby, you must "touch" the ball "down" in the tri zone. It can be assumed that is where the name came from