This is an entertaining book, but it doesn't deserve all that much praise, I'm afraid. First, Armstrong neglects the most important thinkers of the period between 900-200 BCE. By far the most important thinkers are the Greek pre-socratics (Thales, Anaximenes, Zeno, Xenophanes, Heraclitus, Parmenides, Anaxagoras, Democritus, Diogenes, etc), Plato, Aristotle, and the Chinese Warring States era philosophers, like Xunzi, Zhuangzi, and the logicians. These people created our world, from the origins of science to the idea of atomism to the entire realm of secular moral philosophy on which our legal, philosophical, academic, and governmental organisations depend. From them we get the world's first true philosophy - questing, thoughtful, synthetic, and imaginatively unconstrained by tradition.
Armstrong is utterly uninterested in this real intellectual revolution, and instead chooses to focus on the revolution that apparently occurred in religious thinking in the same period. This is spurious in itself; the central thesis of the book is that religion became more humanistic and, well, New Age-y, in this period, instead of consisting entirely of abstract ritual designed solely for the gods. This is bizarre. For a start, no one can truly say what religion looked like before this period. In India, there was no (decipherable) writing before Alexander's invasion in the fourth century BCE, and the vedas, transmitted orally, show a fair amount of humanism as it is. In Babylon and the middle east, the various tales of Gilgamesh (or Bilgames) show a consistent pro-humanity bent against the avaricious, squabbling gods. And in China, the only records we have to go on are the Books of Documents, Songs, Changes, etc, and the Shang oracle bones, none of which show some anti-human inclination. In Egypt, too, stories from as early as 1900 BCE, like the Tale of Sinuhe, show an obvious humanism. There doesn't appear to be anything that required transformation prior to the "Axial Age", and the evidence of a transformation in that supposed age is somewhat spare.
Furthermore, a quick scan of the ethnographic record shows that societies do not progress in some unilineal evolution, and humanistic elements may be found almost anywhere. There is no need for an "Axial Age" to transform religions from being placatory to being humanistic, and no need to go looking in the (sparse and here poorly analysed) data for axial thinkers who just changed everything.
Oh, and the current consensus is that there was no such person as "Lao Tzu". Laozi, the name ascribed to the author of a book named the Daodejing, was almost certainly a composite figure, and the treatise appears to have had several authors, and not just one.
In summary, the reason that an "Axial Age" appears to come about between 900 and 200 BCE is probably due to the proliferation of writing worldwide and the increasing priority and superordinacy of writing over oral transmission in the period. As different views found their ways across the continent (especially with the Alexandrian age), and different positions found themselves in conflict, intellectual problems were created and resolved in new and imaginative ways. This didn't lead only to religious change, but to the development of mathematics, which relied on notation; logic, which relied on complex chains of argument that could not be sustained without writing; and from there, highly developed philosophical positions, drawing on a fertile field of pre-existing writing. That ideas developed in this period is unsurprising, but what is surprising is Armstrong's focus on the silliest and most conservative ideas of the age.
The central position taken by the book is therefore somewhat ridiculous, and this is compounded by the neglect and even disdain shown for the real thinkers of the first millennium BCE: the Greek pre-socratics and their intellectual offspring; the Chinese logicians (like Gongsun Long) and political thinkers (like Han Feizi); and the Indian grammarians (like Panini). Simply put, this is an entertaining book that focuses on some largely non-existent intellectual developments across an arbitrary span of time that happens to coincide with the spread of writing throughout Afro-Eurasia. Writing enabled ideas to spread, and that allowed different viewpoints to clash. While this resulted in many of the most interesting scientific, political, and metaphysical ideas of our species (atomism, recursive grammar, the Hobbesian state, democracy, infinity), Armstrong chooses to ignore these genuinely good ideas in favour of intellectual backwaters.
I have no doubt that it will appeal to people who consider themselves 'spiritual' despite these clear problems.