They dropped a bridge on him. After decades of (sometimes shirtlessly) tangling with the universe's biggest baddies and boning the hottest aliens, Kirk leaves the mortal coil by way of subpar building construction codes.
While watching Star Trek: Generations we knew something was wrong when, during a face-off with the movie's main bad guy with Captain Picard, Kirk tells Picard to hold off the bad guy for him. James T. Kirk passing the chance to punch a dude? That's like a heroin addict saying, "Man, can you shoot up my stash for me? I got an errand to run."
An addiction is an addiction.
So instead Kirk goes to fetch a remote to disable the cloak on a bunch of missiles Soran (the bad guy) was about to launch. The remote just so happens to be on a rickety bridge and, as Kirk manages to make a final act of disabling the cloaking system, the bridge collapses down a cliff, taking Kirk with it.
What Really Happened:
First of all, it's clear that Kirk was shoehorned into the film only because the suits weren't confident they could get people to watch a Kirk-less Star Trek movie (Leonard Nimoy and DeForest Kelley both refused to be in the movie, saying the crew got a perfectly good sendoff in The Undiscovered Country, a film specifically written for that purpose). Then, when the writers were sitting around brainstorming ideas for, you know, what to actually do with him, somebody said, "Why don't we kill Kirk?" (yes, that's literally what they said).