The point of view is third person limited from the perspective of Eckels.
When we talk about point of view, we usually refer to first person or third person. First person point of view has a narrator who talks directly to you, and uses the first person. You can tell because he or she will use the words “I” and “my.” These are first person pronouns.
A third person narrator uses the character’s name and third person pronouns like “he” or “she” to describe the person telling the story. Within third person point of view narration there are two kinds: limited and omniscient. An omniscient narrator knows what all of the characters are thinking and feeling, and a limited narrator follows just one.
This story is told from a third person limited perspective, with Eckels as the narrator. You can tell this right away, by Eckels’s reaction to the sign.
Warm phlegm gathered in Eckels' throat; he swallowed and pushed it down. The muscles around his mouth formed a smile as he put his hand slowly out upon the air, and in that hand waved a check for ten thousand dollars to the man behind the desk.
Eckels is a customer at the time travel office. He is coming to buy a ride in the time machine to shoot dinosaurs. Why he wants to do this I do not know, but he seems a little nervous about it. I would be too. He also seems to be fairly wealthy. It is expensive!
Throughout the story, we know what Eckels is thinking and only Eckels (unless another character is talking and tells us what he is thinking). This is useful because it increases the suspense. We learn about time travel and the butterfly effect just like Eckels.
“…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."
"Why?" asked Eckels.
Eckels will learn why. Oh boy, will he learn why. However, he does not know why and neither does the reader, because the story is told from a third person limited perspective.
If it were third person omniscient, I think Mr. Travis would be thinking, “This idiot is going to alter the course of history by stepping off the path! Why don’t they ever listen? We really should screen these people better.” I imagine it would be something like that.