When you say "butterfly effect," I assume that you mean the idea that one change that seems really unimportant (like a butterfly flapping its wings) can lead (through a bunch of intermediate steps) to a huge change later on (like a big storm). This is really a major point of this story.
In the story, all that Eckels does wrong is to go off the path. While he is off it, he apparently steps on and kills a butterfly. Somehow, millions of years in the future, this change in history has led to a big change in the culture of the United States. Now America has become kind of Nazified when it hadn't been when Eckels' safari went back in time.
When Eckels sees the dead butterfly, he sort of knows (but doesn't want to believe) what he's done:
It fell to the floor, an exquisite thing, a small thing that could upset balances and knock down a line of small dominoes and then big dominoes and then gigantic dominoes, all down the years across Time. Eckels' mind whirled. It couldn't change things. Killing one butterfly couldn't be that important! Could it?