Earlier we saw that before self-awareness hit mankind, women's clothing left the breasts visible. We associate that with primitives in National Geographic, not with the great civilizations of the Bronze Age. But in many ways, those people were still psychological primitives.
From a totally different domain of human life, there's the strange fact that "poetry" did not use meter during the 2nd millennium BC. I put "poetry" in quote marks since I take it to mean something musical and visceral, hence driven by a rhythm or regular beat. Without it, there can still be prose that is lofty in tone, repetitive in word or phrase structure, and ornamented in figures of speech -- but it would still be prose without something special in its prosody.
And just look at the range of languages spoken in major civilizations that were literate and numerate, that produced literary as well as oral texts. Sumerian, the oldest written language, is unrelated to all others, and it didn't use meter. Neither did Akkadian, the oldest recorded Semitic language. Probably not for Ancient Egyptian either, which is distantly related to the Semitic languages by being part of the larger Afro-Asiatic family.  No luck with Hurrian either, yet another completely unrelated language to the others, and spoken in the Mitanni state. 
What about something Indo-European? Those guys were born bards, so surely they must have. Nope, not them either. The oldest attested Indo-European language is Hittite, and their literature didn't use meter. Actually, there's one section of four verses with a regular meter within a larger work -- and that's it. It's more like a brief shift to song-singing mode for whatever purpose, and then returning to (elevated, repetitive, ornamented) prose.
The Rigveda dates to somewhere in the 2nd millennium BC as well, and it does show meter, although since it wasn't written down for so long, I'm not sure how strong the evidence is that the use of meter goes back to the original of 1400 BC (or whenever). Also, it is not a single over-arching work but a vast collection of hymns. So I'm sure for shorter-length compositions, especially in a musical context, meter was used. But not to unify a larger composition. Because the Rigveda is a vast collection, it does not have a meter that's used for just about all of it. Different sections have their own, and choosing from a wide variety of options among the older hymns. Even within such a section, there's plenty of wiggle room around what the rhythm ought to be. 
Mycenaean Greek, the oldest written form of the language, has left no record of their poetry or even "poetry" to study, so we can't say for certain that they didn't use meter. But given how much they psychologically and culturally resemble other groups from before the dawn of self-awareness, I'm pretty sure they didn't use meter in extensive works either.
Meter as a core, pervasive element doesn't really show up until the epics of Homer in the 8th century BC and the Indo-Aryan epic the Mahabharata, from around the middle of the 1st millennium BC. The oldest Chinese poems, from the Classic of Poetry, are in a regular meter and come from the first third of the 1st millennium BC. We can't compare that to earlier stages, as these are the oldest in Chinese. The central books of the Old Testament, also from the mid-1st millennium BC, don't show as much concern with meter as the Indo-European epics, but still more so than their Semitic literary ancestors.
Why the sudden preoccupation with meter, at least relative to the status quo ante? Well, now that people are more self-monitoring, they need new tricks that will lull them into a more spaced-out mindset, so that they can get fully absorbed in the tale-telling, both as a performer or audience member. Following along to a regular beat is one of the most widespread solutions. There's some external pulse, not coming from within your own mind, and following along takes your attention off of your own internal state.
You see that also in contexts of physical group-bonding, like drilling in sports or the military, or group dancing. See McNeill's Keeping Together in Time. Not surprisingly, the mid-1st millennium BC seems to be the starting point for the tradition of military drill, marching in cadence, and so on, from Sun Tzu's China to Classical-era Sparta. People who are not primarily outward-focused need an activity like marching in step to lull the self-aware part of their brain to sleep, and let the animal-like decision-making program take over.
The extensive use of poetic meter would then fall under the range of new tricks that Julian Jaynes saw as attempts to get us back into a bicameral mindset once we'd left that way of thinking and acting. For example, before, people feel some kind of hunch about how to proceed, and go with it, like an animal does; after, they look inside themselves and find uncertain answers, so they turn to increasingly baroque methods of divination. 
So, while bicameral or pre-self-aware people may have had little use for poetic meter, all of us since then need it to get us in the mood, to dial off the spotlight of self-monitoring so that we can get absorbed in the experience. It's like how tapping along to a beat can somewhat take your mind off things, though in a more subtle way so that it doesn't call so much attention to itself as a stylistic device. After all, what would be the point if each swing of the hypnotist's pendulum kept smacking you right on the nose?
 From Foster's Ancient Egyptian Literature: An Anthology (excerpted here):
The major remaining gap in our knowledge of ancient Egyptian poetics concerns prosody. [...] we do not know for sure if it was composed of feet or if it employed some freer means of determining accents and stresses. I would suggest the verse line was analogous to the free verse of Walt Whitman or the modernist American poets. In fact, I think the stylistic texture or flavor of ancient Egyptian poetry can best be described as a fusion of the free-verse rhythms of those poets just mentioned with the rhetorical and structural regularities—the strict attention to patterns of likeness and difference—of Alexander Pope's eighteenth-century heroic couplets (without the end-rhyme or meter). For Sumerian, Akkadian, Hurrian, and Hittite, see A Companion to Ancient Epic.
 From Arnold's Vedic meter in its Historical Development (here):
There are few parts of the verse in which the poets do not consider themselves free at times to depart from the usual rhythms, so that it may perhaps be said that there are no 'rules' of rhythm in the Rigveda. On the other hand, there is no considerable part of the verse in which certain rhythms are not steadily favoured, and others avoided : everywhere there exist metrical preferences. Jaynes suggested that the hunch we used to feel was experienced as an auditory hallucination, a command from our own personal god that we obeyed. Most people who either get really drug-trippily into his work, or those who dismiss it without reading it or reading little of it, seem to latch on to this admittedly wackier part of the entire argument, which is about the shift toward psychological self-awareness.