The poem opens by declaring that the speaker's love, by which he means something more like sexual passion and infatuation, is like a feverish disease. In other words, the speaker, often taken to be Shakespeare himself, has got it bad.
What's more, this love-fever traps the speaker in a destructive cycle. The speaker longs for the exact thing that "nurseth"—feeds or nourishes—this fever: his lover. Indulging in his love might briefly satisfy his longing but, soon enough, that longing would return even more strongly, having been fed "on that which preserve the ill" (that is, having gotten a taste of the very thing that made the speaker so lovesick in the first place). His "love" and his "longing" are thus inseparable, the one feeding on the other like a snake eating its own tail.
The slippery, alliterating /l/ sound ("love," "longing") makes this link clear. Consonance of the same sound appears throughout this quatrain ("still," "ill," "sickly," "please"), imbuing the lines with the speaker's slippery desire.
The polyptoton of "longing" and "longer," meanwhile, emphasizes the fact that this state is ongoing. The speaker's "longing" for "love" makes the situation go on "longer," seemingly without end. It's worth noting, too, that the "fever" has subtly sexual connotations through making the sufferer feel hot (and, perhaps, sending them to bed!). The poem is also alluding to—and subverting—the typical Elizabethan advice to "starve a fever and feed a cold." In failing to "starve" his "fever," the speaker is keeping himself sick.
Love here, then, is not joyful, youthful, and carefree, but rather a kind of parasite destroying its host from within. The poem's meter subtly reflects this idea. As with most sonnets, the poem is written in iambic pentameter—meaning there should be five feet, each with a da-DUM syllable pattern, per line.
But to emphasize the greediness of this sickly (sexual) appetite, line 3 varies swaps a trochee into its first foot, placing the stress on the line's first syllable instead of the second:
Feeding on that which doth preserve the ill,
This metrical variation surprises the reader, emphasizing the eager destructiveness of the speaker's love-disease.
Then, in line 4, the poem uses more consonance and assonance to emphasize just how "sickly" this appetite really is:
Th' uncertain sickly appetite to please.
Try saying this out loud. The hissing /s/ sibilance perhaps suggests saliva, while the /t/ and /p/ sounds have a sharp, biting quality that evokes insatiable hunger.