The first thing Andy does, as he faces forward in the ready position, is establish his grip. Watch his left hand slide down the handle in the top clip. This move not only gets his hands close to each other in a proper grip, but also establishes rhythm to his swing right from the start. Thie sliding of the hand is very common in pro backhands and helps start the overall rhyhtm to the stroke. Marat Safin also does this on every backhand. (Study his backhand here).

Like all great two handers, Andy "stalks" the incoming ball with a slow, gradual unit turn. It's not "the racket" that goes back. Instead Andy slowly turns his shoulders first. And then he slowly takes his arms back as the ball bounces on the court. Note that this is the same formula we see on the forehand side. First shoulder turn. Then arm goes back. During this whole process the racket simply "goes along for the ride" as the torso turns and then the arm extends backwards. Note the gradual, stalking nature of this takeback. Andy is really letting the ball come in towards his body as he slowly turns and takes the arm backwards.

Andy has a very low takeback. Safin, Nalbandian, even Agassi have higher takebacks. But it is important to note that Andy's racket is still angled above his wrists in the takeback. And his racket is above waist height. This will still allow him to use gravity by letting the racket drop.
In the third clip on the right, you can see just how much Andy rotates his shoulders. As he pulls his right arm back, you can see how it actually pulls his shoulder and right lat muscle backwards a bit. This creates a tremendous stretch and coil, not only of the upper body, but of these muscles in particular.

This full stretch also gets the racket and arm extended farther behind the body. Having two hands on the handle limits how far you can take the racket back, so this added stretch lets you maximize the distance of your takeback.

So let's review what Andy does on his takeback. First he doesn't take the racket back quickly. Instead he gradually turns his shoulder first and then extends his arm backwards in what is called a "unit turn". This first move is actually much slower than what club players are used to. The racket does not do anything here - it simply "goes along for the ride" with the shoulders and arm. The racket is cocked above his wrist and is above waist height so that he can take advantage of gravity in the next phase. And finally, his really stretches his right shoulder and right lat to pull the arm and racket even farther back.

Next we will look at the wrist collapse/pull stage of Andy's great two hander.