The math behind the system is mind-explodingly complex, but here's the basic point: Backbreaker does away with canned animations and instead creates unique ones every time. Most games tell the character model, "okay, play this running animation until you get hit, then load one of these ten tackle animations depending upon where you were struck." Backbreaker instead gives the character some basic intelligence and tells them, "Run, and if you get hit, do whatever you can to keep your balance, stay upright and keep running."
This bit of brainpower enables a player in Backbreaker to behave more naturally and dynamically. When hit, a bruising fullback might stumble, but then regain his balance and keep running. If his arm was grabbed, a receiver could keep his legs churning and try to break free. And a tailback might just try to hurdle over one defender but get hit by three others, sending the ball one way and a jumble of bodies somersaulting to the ground in a manner that would put normal humans in a full-body cast.
You should literally never see the same tackle twice, simply because all of the animations are re-computed on the fly each time. And that's to say nothing of how the physics will enable advancements in line play, throwing and catching the ball, and pretty much every other aspect of the game that doesn't involve doing your best to break another man in half.
Thus far, we've only seen a quick, small-scale trailer for Backbreaker (which we're sharing with you below) and some very impressive tech demos - so it remains to be seen how well NaturalMotion can compute the motions of a full field of players without sacrificing in graphics and other areas. If they can make it work, the back most in danger of being broken could be that of Madden itself.