One reason is the fact that the narrator (and there is quite a bit of debate whether it is really Tim O'Brien or not) has killed this man. The vivid description and the imagined history of the victim reinforces the impact of the experience on the narrator. He is responsible for this. In the span of the novel, the narrator goes from being a young kid debating whether to flee to Canada to seeing one of his platoon members killed to now being the one responsible for killing.
Kiowa tries to get the narrator to talk about it, to purge himself of the ordeal, but the narrator, for the most part, remains silent. The story itself, years later, is his attempt to purge himself of the experience and guilt. I think too it shows us that the narrator is not a cold blooded killer. He is not the detached soldier ready to kill when needed. Yes, he kills, but the repercussions linger. This ties into one of the main themes of the novel - the heaviest burdens the soldiers carry are their memories. This is ironic because they can rid themselves of their gear, weapons, rations, and other things, but they cannot get rid of their memories. Physically, these weigh nothing, but mentally and emotionally, they are incredibly heavy burdens.