Travis kills Eckels at the end of the story because Eckels disobeyed the rules of traveling back into time to the dinosaur age—but as we will see, Bradbury implies it is more complicated than that.
Eckels panicked on the safari to the dinosaur age and left the special path constructed to safeguard against any interference with the past that could alter the course of history. In the process, he stepped on and smashed a butterfly: a seemingly minor act, but one that changed the future in profound ways.
We know the future is different when the safari party returns because common English spellings have changed, and more importantly, now Deutscher, not Keith, has won the recent presidential election. As a worker says:
We have got an iron man now, a man who isn't afraid of anyone or anything!"
The piece on information that a would-be dictator has been elected, along with knowing he is responsible for it, anguishes Eckels. Eckels lets out a cry and falls on his knees. This complicates Travis's possible motivations for killing Eckels: the sequence of events suggests that Travis kills Eckels as much out of mercy—pity for his guilt and agony—as for punishment.