Why did the disciples die for their belief in the resurrection?

What the disciples' willingness to die proves is not that the resurrection is true — sincerity is never proof, and plenty have died bravely for beliefs that cannot all be right — but that they genuinely believed they had seen the risen Jesus, something they were uniquely placed to know firsthand. The argument has to be stated carefully: the popular claim that all twelve apostles were martyred outruns the evidence, and as the Christian apologist Sean McDowell has shown, only a few apostolic deaths, like Peter, Paul, and the two Jameses, are firmly attested.

What martyrdom does establish is sincerity, and that matters more than it first seems, because of who these particular people were. A modern martyr dies for a conviction received on trust from others. The original disciples claimed to be eyewitnesses of the very thing in question: that they had seen Jesus alive again. People will die for what they sincerely believe, but they do not, as a rule, knowingly die for what they themselves have fabricated, with nothing to gain. So their willingness to suffer and, for some, to die, closes off the crudest theory, that the whole thing was a deliberate hoax, and confirms that the belief we are trying to explain was genuinely and deeply held.

That leaves the real question open, which is where the interesting conversation begins: what best explains why these first-century Jews became utterly convinced they had seen a crucified man risen bodily from death, a category their own tradition did not even have? Their courage doesn't prove they were right. It proves they weren't lying, and that whatever happened, something happened.