The character is doing something akin to saying grace, but instead of addressing a deity, they are talking to their food. They might thank an animal they killed for its meat, or apologize to it for the kill, justifying it with the kill's necessity for their own survival.
This is usually a sign that the hunter in question is in tune with nature, possibly a Noble Savage. Expect them to make use of all parts of the animal, too.
It's something an Egomaniac Hunter is not likely to do.
Kino apologizes to the rabbit she kills in one episode.
In one chapter of the Keroro Gunsou manga, Natsumi is called in to deal with one of Keroro's invasion schemes that involved exploiting Earthlings' dependency on cattle, which somehow lead to Giroro being turned into a minotaur-esque half-cow, half-Keronian monster and going on a rampage in the Keronian base. When Mom brings home hamburgers for dinner later that day, Natsumi feels inspired to do this trope.
Fullmetal Alchemist: Brotherhood has Ed and Al stuck on an island by their alchemy teacher in a flashback. They catch a rabbit and apologize for killing it, thanking it for extending their lives.
Toriko does this, as part of his belief that life should not be taken for granted.
ElfQuest: In Kings of the Broken Wheel 5, there's a bit of gentle ribbing of Pike, because he always thanks his prey for the meat, even though it can't hear him anymore. 
Sillage has a case of "Sorry, but I need your meat to survive" from Nävis on the very first page, while she's taking aim. The Püntas in Gearing Up have a tradition to this effect as well, to the degree that the yak-like animals walk up to them for slaughter before dying of old age.
"Of all the creatures in the vastness of the universe, there is none like me. I was present at the birth of the universe, and I shall be there at its end. Though I ravage worlds to live, I bear no malice to any living thing. I simply do what I must to survive. And why must Galactus survive? For, no matter how many worlds I devour... how many civilizations I destroy... it is my destiny to one day give back to the universe — infinitely more than I have ever taken from it. So speaks Galactus."
Jondalar, the mate of Ayla in Earth's Children is often described as asking the prey he just killed to confer his thanks to the "Great Earth Mother".
Demonstrated after the deerhunt in Last of the Mohicans, where the hunters thanked the spirit of the deer for its body so that they could live.
In the Redwall books, the Abbey mother would say a prayer to this effect whenever the inhabitants eat a meal of fish.
Happens in one of the Julie of the Wolves books. The hunter also apologizes to the animal whose carcass he was unable to retrieve.
In S. M. Stirling's Emberverse novels, younger members of Clan Mackenzie begin doing this when they hunt.
In the World of Warcraft novel Lord of the Clans, the Orcs thank the Spirits of the Wilds for their prey and promise they won't waste any part of it - its meat for food, fur for clothes, bones for tools.
Inverted: In The Old Man and the Sea the eponymous "Old Man" recalls an incident when he and his apprentice apologised to a fish they'd just caught (although the attitude behind the trope is more or less the same).
The Clan cats in Warrior Cats do this because it's part of their warrior code.
Defied by Tigerclaw, however. He feels he doesn't owe StarClan thanks because he caught the prey on his own.
Wolves from Beyond the Beyond in the Guardians of Ga'Hoole series have a thing at the end of every hunt where they lock eyes with the prey they kill. Doing this is apparently an insurance of the prey getting to the afterlife, and if you can't lock eyes with someone, you basically go to hell, as a not-so-unfortunate bad guy finds out.
The Pahkwa-thanh in the Star Trek Novel Verse. A reptilian race of carnivores who believe their prey animals have souls, they honour the creatures they bring down and kill.
In the Star Trek: Voyager Relaunch, meanwhile, we're introduced to a Klingon tradition of thanking your prey. B'Elanna does so while participating in the "Challenge of Spirit" wilderness survival rite.
"The Sweetest Gift," an episode of the CBS miniseries Christy (based on the book by Catherine Marshall) featured mixed-race Billy Long teaching Creed Allen the importance of thanking a turkey's spirit after shooting it.
In True Blood Amy, a Fantastic Drug addict, blitzed on vampire blood and in the midst of sex with her boyfriend, turns and slurs that they ought to thank their supplier, Friendly Neighborhood Vampire, Eddie "for the gift he's given us." Since Eddie has been kidnapped, chained to a lawn chair in their basement and forced to watch their antics, he replies with a sullen burst of profanity.
In the Call of Cthulhu supplement Terror Australis, adventure "City Beneath The Sands'', Power Boy (who had some relationship with Aboriginal deities) could call animals and ask that they allow themselves to be eaten. If they agreed, he praised their beauty and courage.
Kevin & Kell has this show up at times. At one point Rudy is upbraided for venturing into Jerk Jock territory, and is asked if he's grateful for everyone else involved—his teammates, his coach, and the prey that they all hunt down and kill. Said prey shows up to agree with the point made.
In the Samurai Jack episode "Jack and the Hunters", Aku hires a band of alien hunters who pretty much live for the profession to bring Jack down. They actually succeed in doing so, but not before he gives them the greatest hunt in their careers, so great that they refuse to hand him over to Aku, then let him go and leave him a message thanking him and telling him to "Run free."
This is a real tradition found around the globe, connected with the belief that animals have a spirit/soul that should be honoured. From a psychological perspective, it's a way for people to deal with the cognitive dissonance of life. Hurting creatures is bad, but we still have to eat, and that means inflicting pain and death on something else. We also know that we will eventually die and be eaten in turn. Thus, rituals to try and make sense of it all.
The German inscription in the bottle of Jägermeister: "It is the hunter's honour that he protects and preserves his game, hunts sportsmanlike, and honours the Creator in His creatures". Jägermeister means "master hunter" in German.
There is a joke that goes as follows: A priest is walking through the woods when a bear attacks him. He falls to the ground crying, "Oh, Lord, make this bear a Christian." A moment later he hears growling that sounds like, "May the Lord bless this food that I am about to receive."