Reader response is a type of critical literary interpretation that ignores author intentions as unknown or even unknowable and emphasizes the "I" of the reader, without whom no literature can ever be experienced. While textual considerations are important and limit the validity of various reader response interpretations, the text is not the beginning and ending place as it is with structuralist, formalist, and new criticism literary criticism. Thus the central interpretative element in the critical approach to literature of reader response is the reader's personal subjective reaction/response to the work, provided that cues from the text can validate that reaction/response.
The first step in a valid reader response, however, is understanding the language of the work at hand, in this case, Yeats' "Sailing to Byzantium." Firstly, Byzantium was a great empire that, at it's height in 550 AD, encircled the Mediterranean Sea. It was also the site of the later Eastern sector of the Roman Empire at Byzantium or Constantinople and the even later site of the Orthodox Church that split from The Roman Catholic Church. Yeats seems to refer to Byzantium from its earlier periods through to its history as a religious seat, as he says, "the centre of European civilization and the source of its spiritual philosophy" (eNotes) and the poem says "To the holy city of Byzantium. / ... / sages standing in God's holy fire."
Secondly, the poem says, "perne in a gyre." According to the Irish Literature Companion, a "gyre" is a conical circling movement beginning at the tip, moving toward the broad conical opening and then reversing the circling direction back toward the tip. The word "perne" indicates direction of this conical spin. The poetic speaker, who may or may not be the poet himself, is asking the sages (wise men) of Byzantium to step out from "God's holy fire" and spin in a conical shape ("perne in a gyre") for him.
Thirdly, the second stanza says that "An aged man is but a paltry thing" and that an aged man's only hope lies in his soul singing ("unless / Soul clap its hands and sing, and louder sing"). The poetic speaker then laments that all "singing schools," a metaphor for the philosophies of Western civilization, are busy admiring their own magnificence ("studying / Monuments of its own magnificence"). This connects to the other stanzas because it is for this reason that he escapes and goes to Byzantium: He goes to seek the sages "in God's holy fire;" he goes to ask them to "be the singing-masters of my soul" that he may leave his soul, not as the "paltry thing" but as something made by "Grecian goldsmiths" to be "set upon a golden bough to sing."
A reader response will interpret the significance, meaning, theme, or tropes (metaphors, similes, irony, etc), symbols (e.g., singing-master), myth (e.g., Grecian gods), archetypes (e.g., sage, old man), or allegories of the poem based on the reader's reaction to, impressions of, emotional response to the poem. For instance, one reader response might suggest that while Yeats is using some beautiful imagery to make a significant point about finding renewal outside of Western constraints, his juxtaposition of abrasive language and imagery with elegant language and imagery undermines his meaning and message rather than revealing and underpinning it, and therefore Yeats' stylistic choices undercut the impact of his poem.