The first person view point in this story functions the same way that first person typically functions in most literature.
First, it limits the reader's knowledge of the story. By grounding the story in John's perspective and having it told from only his perspective, the reader experiences the story in the same way that John experiences it. Readers have no extra outside knowledge. As John learns something, the reader learns something.
Second, the first person narration helps to establish the feeling of normal in a world that is definitely not normal feeling to the reader. It's a post apocalyptic world with many superstitions being followed for no announced reason. That's weird. It's nothing like modern day life where answers to questions are just a Google search away. But readers experience John's story, from his perspective. His thoughts and narration treat his circumstance as normal. It forces the reader to quickly adapt. If John says it's normal, it must be.
Third, by having the first person narration, a reader is able to better understand John's internal thought processes and emotions. I think his song ties in well with those feelings. John knows that he isn't supposed to go east to the place of the gods. But he is still going to do it. He's nervous and scared, so he sings his death song. The song is all about bravery and heroic deeds.
"I travel upon the god-roads and am not afraid. E-yah! I have killed the panther, I have killed the fawn!"
The reader gets a sense that John is trying to relieve some of his tension and force himself to be braver. For a while it works. And then the song ends.
"All the same, when I came to the Place of the Gods, I was afraid, afraid."
That shift comes in a single line. Because the reader is reading it from the first person point of view, a reader is really hit hard with the rapid shift in John's emotional state. That kind of thing just doesn't hit readers as hard in third person. It's too distant a writing form.