Byzantium was an ancient Greek city. It’s now Istanbul, Turkey. Before it was Istanbul, it was Constantinople. Byzantium, however, was around from around 670 B.C. to 190 A.D., when it was captured by the Romans. That’s when it became Constantinople.
For the purposes of this poem, however, it’s actually not so important to know all the nitty-gritty details of Turkish history or the decline and fall of the Roman Empire. For all that the poem is set in Byzantium, we don’t actually learn that much about Byzantium. There aren’t any markets or streets or houses or temples or…well, you get the picture. We do hear about the Emperor of Byzantium, but he could be just about anybody. We know that it’s a holy place, but that’s about as much as we get.
OK, so we don’t hear much about the city. But why is it mentioned so many times? It’s in the poem’s title, after all. That’s got to mean something. Here’s our best hunch: since "Sailing to Byzantium" is, after all, a poem about spiritual (and bodily) regeneration, chances are that referring to Byzantium allowed Yeats to draw upon a certain set of cultural references. "Byzantium" becomes a shorthand way to say that the speaker’s entered a mental space that allows him to think through the consequences of mortality. Sure, it helps that there are holy sages around to draw him out of his everyday routine…but that’s not nearly as important as the speaker’s own insistence on finding a way to deal with his own body.