Why did Jack let go of the plank?

While the space on the plank was plenty for two people to sit on it, the plank would have sunk under the combined weight. Since it was on water, it could float only when there was one person on it. If Jack had indeed climbed onto the plank, it would have partially or fully submerged into the cold water. So it was both dangerous and risky - if it didn't sink, then they would be in water and get hypothermia. Hence Jack decided to sacrifice himself so that Rose could stay out of the water and safe on the raft.

Also, James Cameron explains this in an interview

If you watch the movie, and that scene in particular, you'll know that they tried to get 2 on the plank but it began to flip. Jack decided to sacrifice himself rather than put Rose in danger. In the real world, I'm sure they could have eventually worked out the weight distribution so that they both could have climbed aboard without it flipping, but for dramatic purposes they stopped trying after the first time.

To answer the question in the title, Jack never did let go of the plank. He held on to Rose until he was dead (presumably of hypothermia), as Andrew Thompson pointed out.

This is an old question, but I just came across this article so there is now a "direct from the Director" reason:

One question that people ask me a lot about Titanic, and I’m assuming they ask you this a lot, is at the end, why doesn’t Rose make room for Jack on the door?

And the answer is very simple because it says on page 147 [of the script] that Jack dies. Very simple. . . . Obviously it was an artistic choice, the thing was just big enough to hold her, and not big enough to hold him . . . I think it’s all kind of silly, really, that we’re having this discussion 20 years later. But it does show that the film was effective in making Jack so endearing to the audience that it hurts them to see him die. Had he lived, the ending of the film would have been meaningless. . . . The film is about death and separation; he had to die. So whether it was that, or whether a smoke stack fell on him, he was going down. It’s called art, things happen for artistic reasons, not for physics reasons.

Well, you’re usually such a stickler for physics . . .

I am. I was in the water with the piece of wood putting people on it for about two days getting it exactly buoyant enough so that it would support one person with full free-board, meaning that she wasn’t immersed at all in the 28 degree water so that she could survive the three hours it took until the rescue ship got there. [Jack] didn’t know that she was gonna get picked up by a lifeboat an hour later; he was dead anyway. And we very, very finely tuned it to be exactly what you see in the movie because I believed at the time, and still do, that that’s what it would have taken for one person to survive.