What's he saying?
"When I consider every thing that grows / Holds in perfection but a little moment,"
When I think about how every living thing is only perfectly beautiful for a short time,
"That this huge stage presenteth nought but shows / Whereon the stars in secret influence comment;"
We are each like performances on the stage of the world, governed by the workings of the universe;
"When I perceive that men as plants increase, / Cheered and checked even by the self-same sky,"
When I notice how people grow just as plants do,
"Vaunt in their youthful sap, at height decrease, / And wear their brave state out of memory;"
Boastful as youths, then shrinking and dying as they age
"Then the conceit of this inconstant stay / Sets you most rich in youth before my sight,"
Then when I happen to look upon you, you are at your most beautiful,
"Where wasteful Time debateth with decay / To change your day of youth to sullied night,"
But time will eventually make you grow old,
"And all in war with Time for love of you, / As he takes from you, I engraft you new."
But to challenge time, as you grow older I immortalize your youth in my poetry.
Why is he saying it?
Sonnet 15 is one of the "procreation" sonnets of those that are addressed to the fair lord. In it, the speaker contemplates that with time, the object of his poetry will age and lose his beauty. The solution in the final couplet is that the poet will immortalize the youth and beauty he experiences now in the fair lord in his poetry, and thus "engraft you new." The term "engraft" refers to the process of a horticulturist grafting a new slip of wood onto an old root in order to create a new tree.
Sonnet 15 leads into Sonnet 16, also of the "procreation" set. Though Sonnet 15 suggests that immortality can be reached through the poet's "engrafting," Sonnet 16 returns again to the theme of procreation. The final couplet of Sonnet 15 describes how the whole world is "in war with Time for love of you," and Sonnet 16 opens with a plea that the fair lord also defend himself against Time. The speaker calls his rhyme "barren," drawing attention to the fact that although it is one way to immortalize the youth, it does not do as much good as procreation.
In lines 2-3, "this huge stage" is a metaphor for the world. It "presenteth naught but shows," meaning there is no real meaning to what we see in the world; all we see is illusory. Shakespeare likes this metaphor; for instance, we see it in his play As You Like It: "All the world's a stage / And all the men and women merely players." The "secret influence" of the stars refers to the invisible fluid that was thought to emanate from stars, influencing the actions of people on earth.
Lines 11-12 personify Time and decay, as they debate with each other the best way to destroy youth and beauty. But both work together to bring about "sullied night;" night here is described as dirty, in contrast to the "day of youth." Apparenly, the whole world loves the fair lord's beauty, and is "in war with Time for love of you," to protect him against the ravages of time. Of course, this is a losing battle.
The theme of immortality achieved through poetry is Horatian, in that it is not the poet who gains it, but rather the subject of the verse. In Horace's Odes III, 30, 1-5, he writes: exegi monumentum aere perennius, meaning "I have finished a monument more lasting than bronze." In Sonnets 18 and 19, this option for immortality is put forth instead of procreation, which the speaker has been endorsing in the first 17 sonnets.