The context of this passage is that the narrator Xeones is a soldier in the army of Sparta who was severely wounded at the battle of Thermopylae in 480 BCE and who believes that he died and was brought back to life.
The way that I read the passage is that Xeones is describing a near-death experience through the lens of his religious beliefs.
We had reached the river now. We could hear with ears that were no longer ears and see with eyes that were no longer eyes the stream of Lethe
The river Lethe is one of the rivers of the underworld in Greek and Roman mythology. Xeones believes that he, and the other soldiers killed in the battle, became spirits or shades, and therefore, if they were seeing and hearing, it was no longer with their bodily eyes and ears.
and hosts of long-suffering dead whose round beneath the earth was at last drawing to a close.
Some ancient Greeks believed that souls are immortal and, after spending a period of time in the underworld, are reincarnated into new bodies.
They were returning to life, drinking of those waters which would efface all memory of their existence here as shades.
‘Lethe’ means ‘oblivion; forgetfulness’, and in some accounts, the shades of the dead are obliged to drink from this river before being reincarnated, to erase their memory of their time in the underworld.
The reincarnation of souls and the river of forgetfulness do not appear in Homer or Hesiod, and must have been introduced to Greek religion some time after the 7th century BCE. Herodotus, writing in the mid-5th century, says that the doctrine of reincarnation was imported from Egypt:
The Egyptians say that Demeter and Dionysos are rulers of the world below; and the Egyptians are also the first who reported the doctrine that the soul of man is immortal, and that when the body dies, the soul enters into another creature which chances then to be coming to the birth, and when it has gone the round of all the creatures of land and sea and of the air, it enters again into a human body as it comes to the birth; and that it makes this round in a period of three thousand years. This doctrine certain Hellenes adopted, some earlier and some later, as if it were of their own invention, and of these men I know the names but I abstain from recording them.
Herodotus (440 BCE). The Histories 2.123. Translated by G. C. Macaulay (1890).
The earliest mention of a river of forgetfulness, and the most complete account of the doctrine of reincarnation, is in Plato’s telling of the myth of Er, son of Armenius:
All the souls […] marched on in a scorching heat to the plain of Forgetfulness [Lethe], which was a barren waste destitute of trees and verdure; and then towards evening they encamped by the river of Unmindfulness [Ameleta], whose water no vessel can hold; of this they were all obliged to drink a certain quantity, and those who were not saved by wisdom drank more than was necessary; and each one as he drank forgot all things. Now after they had gone to rest, about the middle of the night there was a thunderstorm and earthquake, and then in an instant they were driven upwards in all manner of ways to their birth, like stars shooting. He himself [Er] was hindered from drinking the water. But in what manner or by what means he returned to the body he could not say; only, in the morning, awaking suddenly, he found himself lying on the pyre.
And thus, Glaucon, the tale has been saved and has not perished, and will save us if we are obedient to the word spoken; and we shall pass safely over the river of Forgetfulness [Lethe] and our soul will not be defiled.
Plato (c. 380 BCE). The Republic X. Translated by Benjamin Jowett (1888).
A group of golden funerary tablets from the 4th century BCE, thought to be associated with the Orphic religion, contain instructions for the deceased, describing two pools or springs in the underworld, one for forgetfulness, and one for memory. Richard Janko reconstructed the archetype as follows:
You will find on the right in Hades’ halls a spring, and by it stands a ghostly cypress-tree, where the dead souls descending wash away their lives. Do not even draw nigh this spring. Further on you will find chill water flowing from the pool of Memory: over this stand guardians. They will ask you with keen mind what is your quest in the gloom of (?)deadly Hades. They will ask you for what reason you have come. Tell them the whole truth straight out. Say: “I am the child of Earth and starry Heaven, but of Heaven is my birth: this you know yourselves. I am parched with thirst and perishing: give me quickly chill water flowing from the pool of Memory.” Assuredly the kings of the underworld take pity on you, and will themselves give you water from the spring divine; then you, when you have drunk, traverse the holy path which other initiates and bacchants tread in glory. After that you will rule amongst the other heroes.
Richard Janko (1984). ‘Forgetfulness in the Golden Tablets of Memory’. Classical Quarterly 34:1, p. 99.
Xeones’ description of the underworld is anachronistic for 480 BCE. At that time, the doctrine of reincarnation was a preserve of the mystery cults, and not a feature of general Greek religion. If Xeones had been initiated into the Orphic or Pythagorean mysteries, then this would have had to show up somewhere in the novel, and there’s nothing like that there. However, it’s the prerogative of a historical novelist to extrapolate beyond the evidence.