Although the poem is called "The Sick Rose" a better title might be "the terminally ill rose" because it is really about the rose's death. But the poem isn't just about any old death; it's about a very strange kind of death associated with "love." This odd pairing—love with death—suggests that death is a more complicated matter than we might think, or that something we often assume is unquestionably "good" can have deadly consequences.
The poem describes a complicated relationship between love and death; love in itself doesn't lead to death, only love that is "dark and secret," love that is treated like something to be repressed or not talked about.
Love and death are associated in this poem because the poem wants to suggest the proximity of life and death, the cycle of nature.