Once everyone returns from the safari, what does Eckles find on the bottom of his shoe? What is the significance of this idea?

Upon seeing the gigantic, terrifying Tyrannosaurus Rex, Eckles is overwhelmed with fear and immediately sprints towards the Time Machine. Unfortunately, Eckels leaves the floating Path and accidentally steps on a butterfly. When Travis, Lesperance, and the other hunters arrive back at the Time Machine, Travis notices Eckels's boots and realizes that he has run off the Path. After a brief argument, Travis makes Eckels retrieve the bullets from the deceased dinosaur before they travel back to the present. When they return to the present, Travis and Eckels realize that the trajectory of human history has dramatically changed after they read the Time Safari sign, which is spelled wrong. Eckels then holds up a piece of dirt from his boot and discovers that he stepped on a butterfly. Bradbury writes,

Embedded in the mud, glistening green and gold and black, was a butterfly, very beautiful and very dead (10).

The death of the prehistoric butterfly has significantly altered the trajectory of human history in what is referred to as the "butterfly effect," which is a phenomenon where every living being or organism is connected in a complex relationship. According to the theory, a single occurrence can have a chain reaction and impact every living being in the world. This change can dramatically alter the course of human history, which is what happens when Eckels kills the butterfly.

It is only after the safari has ended that Eckels realizes that he has killed a butterfly. He finds it on the bottom of his shoe, embedded in some mud. Eckels accidentally stepped on the butterfly when he became afraid of the dinosaur and stepped off the path.
This incident is significant because Eckels has rewritten time. Remember that at the beginning of the story, Eckels and the other men were warned of the potential dangers of not listening to the guides and, more importantly, of stepping off the path. The reason for this is that even the tiniest of changes to the past can have a dramatic effect on the future. This idea is made clear in the following quote:

A dead mouse here makes an insect imbalance there, a population disproportion later, a bad harvest further on, a depression, mass starvation, and, finally, a change in social temperament in far-flung countries.

As we see at the end of the story, Eckels' killing of the butterfly has done exactly that: when the men return to the present day, the result of the recent presidential election has changed.