Oedipus kills him after leaving home, upset at hearing the prophecy in which he was told he would kill his father. He runs away from home because he doesn't want to kill the man he thinks is his father (Polybus). So he comes upon a chariot that knocks him off the road, and he gets so mad, he kills all of the people in it, including Laios. The exact line from the book ( Dudley Fitts and Robert Fitzgerald translation) is : "He was paid back, and more! Swinging my club in this right hand I knocked him out of his car, and he rolled on the ground. I killed him." What isn't clear is whether he killed him by knocking him over the head, which caused him to become unconcious and die, or whether he fell out of the car and then Oedipus killed him. But that is in a nutshell, how Laios died. Hope that helped!
