One of the things that struck me as odd, when it first came up, was the compulsion the Greeks seemed to have felt, with stripping people they killed of their arms and armor. It seems a bit odd that warriors were so obsessed with immediately getting their hands on their opponent's swag that in several cases they were killed while doing it. The narrative remarks on this, but not really in a way that seems critical of the practice overall; just the poor timing.
Although the Great Books technique discourages checking "outside the text" for explanatory notes, I don't entirely adhere to that. I agree that it can take the reader out of the flow of a text, but I don't think it's the end of the world if it doesn't happen too often. There's an advantage, of course, for someone who already knows what kleos is all about, which could become an unfair advantage if others are prevented from finding out. That wasn't me, so I looked it up.
In addition to the fact that armor and weapons represent some of the most important (and probably the most portable) accumulated wealth of these bronze-age cultures, it was apparently very important in the ancient Mediterranean world not only to be brave and win, but to be seen doing so. The fighting style in these battles seems to be that "heroes" took turns rushing to the front line and engaging in what almost amounted to duels with their peers from the other side. Peasant conscripts didn't take on Hector or Diomedes. This may not represent how battles really went during this period; it reminds me of the conceit in modern martial arts movies where everybody hangs back and the protagonist fights a series of one-on-ones instead of being overwhelmed by superior numbers (or simply shot).
Along with this heroic nearly-dueling style of warfare (punctuated by the occasional actual duel, as between Paris and Menelaus or Achilles and Hector) is the imperative of stripping the corpse as quickly and publicly as possible. Kleos is generally translated as glory, but its root is the verb that means "to hear", so there's a tight linkage between honor and being talked (or sung) about. I was surprised that this imperative was so universal that even the god of war, Ares, was obsessed with collecting the armor of mortals he killed (although there may be some ironic commentary going on here too, since Ares isn't depicted as one of the more self-aware or intelligent of the gods). Not only does the ethic of the time (at least for warfare) seem to be incredibly zero-sum, but it's also obsessively performative. You become a hero by defeating an opponent of similar rank and then stripping him of his honor publicly. I think it's unfortunate that things haven't changed that much, in some aspects of contemporary "culture". It's also pretty cool that such an ancient story is so resonant and relevant today.