The first thing we see is that in 2012 Donaldson looks like a rookie. He just has the look. Moving on. At first glance, there doesn't seem to be much difference. You could simply stay it was just a matter of time and practice, maybe confidence. But let's look really closely at Donaldson as he plants his front foot. Notice how he has what some would call a "hitch" with the hands in the 2013 swing compared to the 2012 swing ? (This movement has actually gotten more pronounced as the year has progressed from this video of the early part of the season.) Instead of just launching his hands straight at the ball as in 2012, he is putting them in position to allow his legs and torso to get some bat speed generated.
Here's is a shot of the frames I am focusing on. Notice how in the last frame his hands work down toward the ground in 2013, instead of straight at the ball in 2012. He isn't being "direct to the ball."
By allowing his legs and torso to work more before using his arms, he has a slightly faster swing - about half a frame at 30 fps. The quickness allows him to improve on the -13.5 weighted value he achieved on fastballs in 2012 (according to fangraphs.com). In 2013, he is up to 9.1 on fastballs. (As a side note, Donaldson is positive on everything but change-ups, and only at -0.5. That's friggin awesome.)
Do I know if this swing change is the result of A's coaching or even a conscious thought, no. But I do know that trying to "keep the hands quiet" isn't a great cue for everybody, including Josh Donaldson.
(BTW, I called Donaldson having a tremendous year on May 9th. http://www.eastbayhittinginstruction.com/top-hand.html)