Here's how I have heard it explained:
A chief ingredient in the recipe for firearms accuracy is repeatability. Yes, you need a good barrel, stiff receiver, fast lock time crisp trigger and a precise sighting system as well. However, when you fire, everything in that rifle that has mass (ie everything) is going to move as a wave of energy passes through the rifle.
The trick is to get all that motion to be minimal and to have all of the parts of the rifle come back to exactly the same place every time.
In a bolt action it's relatively simple: you full bed / free float the barrel in a way that the "wave" moves through the rifle clean with everything returning to the same place. You tune your load / adjust your muzzle weight so the bullet leaves the barrel at the same point in the wave every time.
In a Semi auto, it's a bit more complex. you have reciprocating parts, usually moving on different vectors than the bullet and in an unsynchronized way. That tends to flap all of the rifle parts around so that they don't come back to exactly the same place each time.
What the AR does is first, reduce the number of reciprocating parts. Second, lighten each one as much as is practical. Third, move these little parts in a straight line in a synchronized way.
By using a gas tube instead of an operating rod and piston along the barrel, you eliminate a large mass that normally slams back and forth torquing the barrel and action all over the place each cycle. next make the bolt, and bolt carrier lighter than is normally possible.
Next Take a stiff milled forging of a receiver, screw the barrel in to it, draw a straight line through this and place all of the reciprocating parts on that line. The barrel lines up with the bolt which is ultra light, and is also the piston (yes an AR has a piston it's just been moved so it's in a straight line with all of the other reciprocating parts) that goes into a concentric bolt carrier. The bolt carrier is ultra light so right behind it is a buffer weight backed up by the action spring. All in a straight line. Some genius.
Now comes the REAL genius. Ordinarily, the bolt would be to light to function without tearing itself up rather quickly from the unlocking forces on those little bolt lugs. But, in the first few microseconds of fire, the gas moves down the tube into the bolt carrier filling the piston chamber evenly and as the bolt carrier begins to move back, the bolt moves FORWARD a small amount unloading the forces on it before it rotates and unlocks and the entire bolt and carrier begin to move back. As the bolt carrier is light the initial motion is less energetic and violent than if you combined the mass of recoil buffer and the bolt carrier as in a traditional semi auto. this stretches out the cycle smoothing it and reducing the effect of the reciprocating parts.
Also, since all reciprocation is concentric to the bore, you no longer need rails milled into or spot welded to the receiver to force everything to move in a straight line.
Here is a long stroke gas piston rifle flapping around
http://www.youtube.com/watch?v=xeeeFxA_9nA&feature=related
Here are the M1-A and M1 Garand
http://www.youtube.com/watch?v=SgldaFmgPZo Watch the barrel motion.
Here's an M16 head to head with an AK
http://www.youtube.com/watch?v=G6BpI3xD6h0