In the movie The Search for Spock it is mentioned that Spock's rapid aging and the Genesis planet's turbulent nature are side effects of protomatter being used in the creation of the Genesis device. Considering the planet's end result (destruction) why didn't Spock's new body suffer a similar fate of a rapid death?
The Genesis team didn't have the problem of runaway catastrophic evolution in the cave (where Khan left them to die), so perhaps the runaway effect was caused by too high a concentration of protomatter. Somewhere between the size of the cave and the size of a planet a tipping point was reached. When Spock was moved a large distance from the concentration of protomatter on the planet, his runaway aging stopped.
The planet was based on the Genesis probe. It was created out of, essentially, nothing (other than, maybe, the gas particles from the nebula that was there in the battle with Khan, but seemed to disappear later).
Spock, however, was created from a Vulcan body, that still had DNA and cells and mass. Remember, the original Spock's body was never found. His robe was, but his body was not. Young Spock was not a part of the Genesis planet and was not actually created with it. The planet was already there when the torpedo casing (his coffin) landed on it.
It's not clear just how much of a rebirth Spock received. Whether a living being started growing directly from his DNA, or whether his original body was essentially returned to an earlier, more pristine version of the original, without cancer cells or scars or anything else that he would have gained along the way. (Remember, also, his robe was found folded and not just left in a pile.)
But the young Spock that we do see does grow from a young, smaller body, to a larger one. How the additional mass was added to his body so quickly is never made clear, but the only source for that would be from the Genesis planet. It's likely he would have had to eat whatever he could and would be ravenous before and during growth spurts.
When he's removed from the planet (also when it blows up), he stops aging rapidly (which, coincidentally means he stops aging when his body is at almost the exact same age as when he died). There are two things that happen at that point: 1) He no longer has food or material from the unstable planet to eat (and it may be that food that fueled his growth and rapid aging), and 2) If there was any matrix (as David referred to it), it was destroyed with the planet, which would also stop the rapid aging.
So there are two likely reasons he would have stopped aging.
More than likely the removal of Spock from the Genesis planet played a big role in the halting of the aging effect. I would suspect that once he was out of range of the planet and the effects of the Genesis project the symptoms would then also begin to stop as well. I also read somewhere that another possibility that combined with what I've just said, when Spock was beamed up to the ship, the transporter may possibly not be able to detect the Genesis effect and therefore doesn't include it when transporting matter or rematerializing it. If this was the case then the Genesis effect and everything about it would have been left behind once they were off the planet and out of range.