The climax of a story is the turning point. It's the point where everything changes. The climax of this story occurs once Eckels and the safari group return to the present. They realize that the timeline has been changed in a significant way, and the cause of it all is the death of a single butterfly. It is the butterfly that Eckels stepped on while he was trying to get back to the time machine. It's at this point that the readers are reminded of Travis's words earlier in the story.
"Step on a mouse and you crush the Pyramids, Step on a mouse and you leave your print, like a Grand Canyon, across Eternity. Queen Elizabeth might never be born, Washington might not cross the Delaware, there might never be a United States at all. So be careful. Stay on the Path. Never step off!"
I have seen other climax points for this story defended too. One such possible climax location is when Travis realizes that Eckels has left the path.
"He ran off the Path. That ruins us! … We guarantee no one leaves the Path. He left it. Oh, the fool! I'll have to report to the government. They might revoke our license to travel. Who knows what he's done to Time, to History!"
At this point, nobody knows that the present has been changed. Nobody knows that Eckels has stepped on the butterfly. All that we know is that Eckels has likely caused some kind of dramatic change. If this part is the climax, then the realization that Eckels stepped on the butterfly and changed the entire present would be considered falling action.
I do think a third possible climax location for this story is when Eckels faces the dinosaur. If the climax of a story is considered a significant turning point in the course of events, then I think the dinosaur scene can make a legitimate case for being the climax. Before Eckels faces the dinosaur, the reader has confidence in his abilities. He's annoying, but he is an experienced hunter and talks brave; however, when the dinosaur shows up, everything changes. It's that moment in the story that causes everything else to go wrongly. If Eckels never faces the dinosaur, he never gets scared. If he's not scared, he doesn't leave the path, step on the butterfly, and change history. Climax scenes also tend to be tense and exciting. Returning to the present to find that everything has changed is tense, but I don't feel it's as tense as the dinosaur encounter.