! None of what happened in the movie is real
Cole is in fact a schizophrenic patient, but the only parts of the film we see happen in reality are the scenes where he is in the hospital talking to psychiatrists. Both the "present" and future worlds were artificially constructed in his head. As a child, he was traumatized by an airport shooting, which in turn led to him constructing a world in his head. The Psychologist, Dr. Railly is real, and has been helping Cole for some time, but she is only going along with his delusions and occasionally having a bit of fun along the way (being able to leave a prank call about the Twelve Monkeys). Ultimately, Railly's help has started to bring Cole back into reality, but he is still under the delusion that he is a time traveller and requires her assistance, but occasionally he experiences a temporary relapse (his apparent returns to his own time). The climax is ultimately Cole coming to face the source of his delusions by experiencing the shooting again and confronting his delusions which, though not shown in the film, finally bring him back into the true reality.