The first line is comprised of a tri-colon (a list of three) accumulating three parts of the process of ageing which combine to give the sense of someone approaching death - 'old', 'grey' and 'full of sleep' and are meant as a juxtaposition (deliberate poetic contrast) to the remembrance of a younger self that comes in the following lines.
The phrase in the second line of 'nodding by the fire' gives a metonymic image of this contrast, the idea of the fire is a metonym for youth and vitality - the inner fire of memory of a youth now past can be re-kindled by reaching for a 'book' (indeed, 'this' book - which implies the book in which this poem is contained) in which the old woman can read a description of herself as a young and beautiful woman, thus re-kindling in her mind at least the image of herself as beautiful. The woman in the poem is most likely Maud Gonne, Yeats's companion and sometime lover, who he repeatedly wrote about in poems such as 'The Wild Swans at Coole' (and proposed to many times, without success, before marrying Georgie Hyde-Lees). The line refers to the famous poetic device of imagining being described at a young age in a literary work as giving one immortality (as one does not age within the pages of a book), an idea expressed in early works such as Shakespeare's Sonnet 18 in which the poet describes how in 'eternal lines to time thou grows't'.
The final lines of the first verse create a shift in time, describing the earlier beauty of the old woman when young and the 'soft look/Your eyes had once'. At one level, this is poignant in its description of a lost beauty but at another implies that the eyes have hardened with age, the implication of this description that the woman has got harder and less approachable with age, a hidden reproach on the poet's part that one might read as resentment at his own rejection.
The second stanza creates a shift of time to a remembrance of the old woman in her youth as a celebrated beauty who 'many loved', again a part of the biography of Maud Gonne, as RF Foster describes in his excellent biography of Yeats. However, he also comments in the second line of the stanza about the possible insincerity of some of the love she was showered with as 'love false and true' and one wonders whether there is some self-reproach in his comment, that perhaps he wonders whether he too was dazzled by her beauty into a form of 'false' love. However, this confusion is resolved in the final two lines, perhaps, with the revelation of a single loyal lover who loved 'the pilgrim soul in you'. This rather oblique phrase may refer at one level to a man who sees beneath the skin-deep beauty of the younger woman (which, as we know from the first stanza, has faded with age) and instead loves a 'soul' loyally dedicated to one thing. This might refer particularly to Gonne's staunch belief in Irish republicanism, held throughout her life, the soul in this case being 'pilgrim' as she was English, rather than Irish by descent. Her 'sorrows' may refer to the abusive relationship she endured at the hands of her first husband, John MacBride, another staunch republican who was executed by the English in the aftermath of the Easter 1916 Rising described by Yeats in 'Easter 1916'.
The personification of love in the last stanza of the poem is prefigured by the old woman being divided from the fire, the ethereal symbol of life force, when she is 'bending down beside the glowing bars' (the separation emphasised by the hard 'b' alliteration). Love appears to have 'fled' and paces wild amongst the 'mountains', free while old age constrains the old woman, but for the fact that her beauty as a younger woman has been captured in the pages of the poet's verse.