To a certain extent, this is an opinion based question. Feel free to state why you think Travis is annoyed with Eckels, and then defend it.
In my opinion, Travis is annoyed with Eckels because Eckels is annoying. Stick with me. I relate much better to Travis than I do with Eckels. I find Eckels annoying, so I figure that Travis finds him annoying for all of the same reasons that I find Eckels annoying.
Eckels is annoying because he is cocky and pompous. Eckels is annoying from the very moment that he walks into the Time Safari, Inc. office and waves around his check. I don't mind people being generous with their money, but people that openly flaunt their wealth are annoying. Eckels waves his check in the air as if trying to prove to everybody else that he is better than they are because he can write a big check. That's annoying.
Next, Eckels is annoying because he's selfish. Eckels is part of a safari expedition into the past. There are other paying customers, yet Eckels doesn't seem to understand that. He thinks that the entire expedition is there to serve his hunting desire. He thinks that his big check has gained him exclusive rights to shooting a dinosaur.
"Shooting my dinosaur," Eckels finished it for him.
Third, Eckels is annoying because he's foolish and quick tempered. Before heading into the past, the safari company clearly and calmly explains that the expedition is dangerous. The company asks Eckels to sign a consent form. Instead of taking to heart the severity of the trip that he is about to undertake, Eckels gets angry at the employee.
Eckels flushed angrily. "Trying to scare me!"
Eckels could have said "okay" and signed the document, but he gets accusatory instead.
Fourth, Eckels is annoying because he has no respect for the rules that have been put into place. Travis very calmly and clearly explains the dangers of messing around with events in the past. Instead of simply agreeing with the rules, Eckels has the gall to tell Travis that the rules are overly cautious. He doesn't see the problem with killing a single mouse that might lead to the deaths of billions of mice.
"And all the families of the families of the families of that one mouse! With a stamp of your foot, you annihilate first one, then a dozen, then a thousand, a million, a billion possible mice!"
"So they're dead," said Eckels. "So what?"
The fact that Eckels is so concerned about getting what he believes is his leads him to not care about the destruction of billions of living animals. That's rude, and rude is annoying.
Fifth, Eckels is annoying because he doesn't understand basic gun safety. While walking along the path, Eckels points and aims his gun at imaginary targets like the gun is a toy.
Eckels, balanced on the narrow path, aimed his rifle playfully.
That's just ridiculous and unprofessional.
Sixth, Eckels is annoying because he can't follow simple directions. Once the dinosaur shows up, Eckels gets scared. Travis quietly and calmly tells Eckels to be quiet and head back to the time machine. Unfortunately, Eckels can't follow those simple directions. He keeps talking and standing there despite Travis telling Eckels numerous times to be quiet and head back to the time machine.