star trek - Why didn't Spock die in The Search for Spock?

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.