There is no one answer: a key part of this poem's appeal is its ambiguity.
On the surface, it seems a poetic description of a rose flower sickening and dying due to a parasitic infection. However, the opening lines make it clear it should not be read literally: no "worm" is "invisible", nor does it "fly". So how can we interpret the metaphor?
First, what is a "worm"? It could be a parasite or disease, sure, but it's also another word for a snake. The rose in its bed might represent the state of nature, the garden of Eden. In this interpretation, it is a poem about how "secret love" made "sick" and "destroy"[ed] the "rose" of humanity's state of grace in the Biblical narrative.
In the Bible, we also learn of demons which tempt humanity into further sin in order to ensnare their souls. According to medieval legend, a particular class of demon is the incubus or succubus which visits sleeping victims, offering to slake their lustful appetites. In this second interpretation, the "worm" is a symbol of corruption. Like a demon it "flies in the night" and shares its "dark secret love" with the mortal "rose" which it wishes to "destroy".
The forbidden fruit of Eden and the demonic temptations are, of course, sex. This is the third interpretation. The "bed" here is one of sexual pleasure, as represented by the "crimson joy" of the rose and the "howling storm" of orgasm. Red is a colour associated with passion and with life. Worms, however, are both phallic in form - hinting at the potential of violence - and a symbol of death. Here, the "rose" of passion is "sick" with the knowledge of mortality and ageing which gradually eat away at beauty and sexual pleasure.
Fourth, we can interpret the poem as one about relationships in a less directly sexual manner. The "worm" that "flies by night" is suggestive of the "invisible" fear and negativity we sometimes experience during the night, especially during a "howling storm". It's a sad but realistic way for poisonous thoughts to enter a relationship, perhaps the relationship to the person you share your "bed" with, your "rose", poisonous thoughts which might come to "destroy" that love.
Sticking with emotion, a fifth interpretation is offered via another Blake poem in the same volume, A Poison Tree. This is a much less metaphorical work that tells a fairly straightforward story about how nurturing anger can lead to brutal, even deadly consequences. Here the "howling storm" is emotional turmoil and the "worm" the seed of bitterness or anger which slowly "destroy"[s] the "rose" of a sound and positive personality.
One of the most striking things about this poem is that almost all these interpretations are interlinked. Biblical sin is often sexual, sex combines with life and death in an endless cycle, society shames sexual misconduct leading to shame and anger. This deep interlinking of themes makes the poem almost impossible to pin down to a single interpretation. Its strength is that by hinting vaguely about sex and relationships, it retains the potential to offer wisdom across that whole aspect of the human experience.
