Eckels travels with Time Safari, Inc. because he wants to go back in time to shoot a dinosaur. 

Eckels is an expert hunter looking for a new challenge.  He spends a lot of money to hire the time machine so that he can go back to shoot a dinosaur.  Obviously, it is no longer possible to shoot a dinosaur on Earth since they are extinct.  Therefore this company offers brave hunters the opportunity to shoot an animal far bigger than any alive today. 

Eckels is told that his safety is not guaranteed and that if he does not follow directions he will have to pay an exorbitant fine.  The discussion with his guides is about a recent election, but they tell him not to worry about that.  He is going back in time well before the election ever happened. 

“All you got to worry about is-" 

"Shooting my dinosaur," Eckels finished it for him. 

"A Tyrannosaurus Rex. The Tyrant Lizard, the most incredible monster in history. Sign this release. Anything happens to you, we're not responsible. Those dinosaurs are hungry." 

Eckels is told not to step off a special path, and not to shoot anything else.  It is very important that he not change the future.  Any little thing he does could do it, and that is why the rules are so strict.  When Eckels comes to the point of actually shooting the dinosaur, he panicks. 

It ran with a gliding ballet step, far too poised and balanced for its ten tons. It moved into a sunlit area warily, its beautifully reptilian hands feeling the air.

"Why, why," Eckels twitched his mouth. "It could reach up and grab the moon."

He is so frightened that he is unable to actually shoot.  His guides end up doing it and during the commotion that follows he accidentally steps on and kills a butterfly.  When they return to the present everything has changed, including the outcome of the election.  Travis shoots Eckels.