It didn't. The very idea that Ancient Greeks didn't have an inner self-consciouness (as proposed in the book) is invalid (even the article says so).
That said, what ancient civilizations didn't have, and was developed culturally and through time, is the kind of complex self-introspection we have now.
In a way the Ancient Greeks (and other people) were more like James Stewart (straightforward and simple) than Woody Allen or Orson Wells (full of clashing thoughts, ideas about guilt, sin, self-introspection etc). Their inner thoughts they externalized to some degree (which is also the basis behind the book). E.g. guilt was seen as external entities "haunting you" (e.g. "furies" in ancient greek tradegy). Of course in a degree they understood it was coming from them, but they didn't have a fully developed framework to talk and introspect those feelings.
A lof of those ideas only developed fully in the 2.5 centuries since then, and Christianism played some role in that, as did religions like Zen Budhism etc in the East, that re-examined and explored lots of things about the "inner self".
That would be "millenia".
1. I recall that the events described in the Illiad did not take place in Homer's lifetime, but much earlier.
2. The Bahagavad Gita, which is one of the texts the previous poster alluded to, is likewise a tale of events that allegedly unfolded long before its writing (some 5000 years ago allegedly).
Unfortunately this brings us to a bit of a circular argument, that recorded history began because consciousness began, and vice versa.
(By no means am I claiming that these areas are entirely isolated from each other, but note the bandwidth of the cultural communication from the travel between them is so low that the net effect is that the various areas of the ancient world were basically unaware of each other. Even if some travelers occasionally made the hikes they didn't amount to much at the civilizational scale.)
And I find it far more parsimonious an explanation that this is just a crazy idea from a too-close reading of a work of literature that thematically chooses to be about "the gods" to imbue the work with mythic power, just the fact our culture has Star Trek does not mean that we have warp drive or that Gene Roddenberry's vision of happy coexistence has been realized. A fun idea, a great idea to build some sci-fi on, but not really a serious idea supported by the totality of the evidence we have from history, in which humans have been musing about the nature of consciousness for basically all of recorded history.