Shakespeare's subject in Sonnet 63 is the transience of life, the power of memory, and the permanence of poetry.
In the first four-and-a-half lines, for example, the poet prepares himself for the point at which old age begins to change his lover:
Against my love shall be, as I am now,/With Time's injurious hand crush'd and o'er-worn;/When hours have drain'd his blood and fill'd his brow/With lines and wrinkles
First, the poet describes himself as a victim of time, which has worn him out, and he is developing a defense against the time when his lover is also "crush'd and o'er-worn." Shakespeare skilfully uses balanced language to create images in his reader's mind: drained his blood/filled his brow with wrinkles. As usual with Shakespeare, he takes the abstract concept of aging and makes it real with imagery.
Shakespeare uses the metaphor of age as a journey in the second half of line four and line five when he notes that his lover's "youthful morn/Hath travell'd on to age's steepy night," a journey that ends with his lover's physical beauties disappearing (ll. 5-6)--literally "stealing" youth ("his spring") from his lover.
Beginning with line nine, Shakespeare discusses the measures he will take to blunt
. . . age's cruel knife,/That he shall never cut from memory/My sweet love's beauty, though my lover's life. . . .
In other words, even when his lover is dead, the poet's memory will prevail and conserve his love's youthful beauty.
The couplet (ll. 13-14) makes the argument that the poem--"these black lines"--will live on, and in lving on, will always present the poet's lover "in them still green," that is, always youthful and beautiful. In essence, then, the lover lives in the memory, which is then recorded by the poem, and the poem is what remains.