The story
Usually said to be from the maximum-speed state of a ball governor, whose balls are swung out to the near-horizontal position. Neither balls-out nor balls to the wall is connected with the vulgar sense of balls (“testicles”) except connotationally via folk etymology. (In that vein of apparent vulgar innuendo, see also rock out with one's cock out.)
Etymology from Wiktionary, CC BY-SA 4.0.