Eckels is an arrogant person; he is used to being in charge, yet he lacks self knowledge. He doesn't consider the serious ramifications of time travel. It's simply another experience he can consume.
He therefore underestimates the risk of traveling back to the dinosaur age. In fact, he gets angry when the man behind the desk suggests he might be eaten by a dinosaur as Eckels signs the release form that acknowledges the company is not responsible for what happens to him:
Eckels flushed angrily. “Trying to scare me!”
Eckels also has romantic ideas about the past. He is taken in by the time travel company's advertising, which he remembers as
everything fly back to seed, flee death, rush down to their beginnings, suns rise in western skies and set in glorious easts, moons eat themselves opposite to the custom, all and everything cupping one in another like Chinese boxes
Eckels likes the idea that his money can buy him the right to shoot an animal from the past and initially has little understanding as Travis tries to explain to him the dangers of changing even the smallest event from long ago.
Eckels, used to being in charge, doesn't like to be told what to do. He flushes (a sign of his anger) back in the dinosaur age when Travis tells him to put down the gun he is aiming playfully. Eckels tries to regain control by asserting possession, asking Travis "Where's our Tyrannosaurus?”
When he sees it, he decides for himself that it can't be killed, showing he is used to consulting his own opinion and nobody else's. If he thinks something, it must be true. He's not used to being deferential or humble:
“It can’t be killed.” Eckels pronounced this verdict quietly, as if there could be no argument. He had weighed the evidence and this was his considered opinion.
Of course, he is wrong. He also overestimates himself: the Tyrannosaurus frightens him much more than he expected, and this causes him to leave the path and enter the jungle.
His reaction to the dinosaur and his mistake in killing a butterfly humbles him to the point he apologizes, but it is too late. His arrogance has already changed history.