Eckels is the man who changes the present. . . by changing the past . . . because as George Orwell wrote, "whoever controls the past, controls the present."
The first thing to know about this story is that time travel into the past is possible. In fact, it's common enough where it has become a tourism industry. The story begins when Eckels walks into the offices of Time Safari Inc. He wants to go back in time to hunt and kill a dinosaur. As Eckels learns about the rules, so does the reader. The time travel company will take him back in time to the time of dinosaurs. The time travel company also provides a guide and a really cool floating sidewalk thing that doesn't touch anything in the past.
“And that,” he said, “is the Path, laid by Time Safari for your use. It floats six inches above the earth. Doesn’t touch so much as one grass blade, flower, or tree. It’s an anti-gravity metal. Its purpose is to keep you from touching this world of the Past in any way. Stay on the Path. Don’t go off it. I repeat. Don’t go off. For any reason! If you fall off, there’s a penalty. And don’t shoot any animal we don’t okay.
The guide is to ensure that Eckels kills the correct dinosaur. The dino in question is a dinosaur that is about to die anyway, so Eckels actions won't actually change anything in the past that wasn't already going to happen soon after.
“Right,” said Lesperance. “I track them through their entire existence, noting which of them lives longest. Very few. How many times they mate. Not often. Life’s short. When I find one that’s going to die when a tree falls on him, or one that drowns in a tar pit, I note the exact hour, minute, and second. I shoot a paint bomb. It leaves a red patch on his side. We can’t miss it. Then I correlate our arrival in the Past so that we meet the Monster not more than two minutes before he would have died anyway. This way, we kill only animals with no future, that are never going to mate again. You see how careful we are?”
The time travel company doesn't want to cause a ripple effect through time by doing something in the past that would change thousands of generations of life.
“say we accidentally kill one mouse here. That means all the future families of this one particular mouse are destroyed, right?”
Unfortunately for the story, Eckels gets scared of the dinosaur that he is supposed to shoot. He accidentally steps off of the path and kills a butterfly. When everybody returns to the present, the entire United States is a militarily controlled dictatorship.
Eckels, not looking back, walked blindly to the edge of the Path, his gun limp in his arms, stepped off the Path, and walked, not knowing it, in the jungle. His feet sank into green moss. . . Eckels felt himself fall into a chair. He fumbled crazily at the thick slime on his boots. He held up a clod of dirt, trembling, “No, it can’t be. Not a little thing like that. No!”
Embedded in the mud, glistening green and gold and black, was a butterfly, very beautiful and very dead."
It's Eckels who helped create the now familiar phrase "The Butterfly Effect." The phrase is referencing how one small change can have huge consequences.