This seems to me to start off as a poem intended to say something powerful about nature and the fact that man is a predator, but the vocabulary used negates that idea - the words "trundle" and "bundle" are quite colloquial and make it sound as if Auden is poking fun.
This is also suggested in stanza 3, when he refers to the poem as "deathless verse", which is a bit of a cliche. It used to be a powerful phrase when it was first used to describe, say, Homer's epics, but is now used tongue in cheek, for instance to describe the really bad verse by McGonagall in the Tay bridge Disaster.
Taking the poem stanza by stanza - stanza 1 simply describes the way a hunter will kill an animal or bird and bring it home to be cooked. There is a feeling that the poet is sorry for the animal, calling it some "he or she" to give it a more human shape, and referring to the hunter as "some example of our tribe" as if he is an anthropologist and the hunter a subject for study.
Stanza 2 compares lovers, man and woman, and tells us how like hunter and hunted they are, although he gives the woman more power in the man's eyes by suggesting that her "witch's heart" is an "oven" just as in Hansel and Gretel, she intends to cook and eat him.
There is an interesting idea here, a powerful point to be made about hunting and about the relationship between the sexes, as well as about the need for humans to hunt and kill. However, Auden seems to run out of steam, whether intentionally (as suggested by the fact that he uses colloquial language from the start) or because he just hasn't got the words to make it work. He refers to the hard chair to tell the reader he's feeling physically uncomfortable and needs to move, he refers to the poem as "deathless prose" so putting himself down, and then tells us he's having a meal of fish, another living thing, so he's no better than the hunter he's been looking at, and is human too.