In "I heard a Fly buzz—when I died—" the speaker describes the moment in which she transitions from life to death. In the opening stanza, the speaker emphasizes the "stillness" and quietness of death. The moment is so still and so quiet that she can hear "a Fly buzz." The speaker feels that this still, quiet moment is "like the Stillness in the Air / Between the Heaves of Storm." In other words, she feels as if this moment is the quiet before the breaking of the storm. This is a rather ominous simile, as it suggests that death itself will be loud, chaotic, and aggressive, just like a storm. We might infer, therefore, that the speaker is afraid of what death might bring.
In the second stanza, the speaker describes the mourners who are there to witness her passing, and she anticipates being with Christ ("the King") once she has finally left her life behind. This peaceful moment in which the speaker accepts death is interrupted in the third stanza when the speaker once again hears the buzzing of the fly. The fly here serves to undercut the seriousness of death. The fly to some extent trivializes the speaker's moment of passing, and perhaps in this way Dickinson means to suggest to the reader that death is not something that we should be afraid of.
In the fourth and final stanza, the speaker describes the awkward, ungraceful movement of the fly. It's "uncertain stumbling" perhaps contributes to the sense that death is, in this poem, being undercut or trivialized. The "uncertain stumbling" of the fly perhaps also is meant to echo the uncertainty of the speaker as she leaves her life and is about to enter the unknown realm of death.
In the first line of the final stanza, the speaker describes the "buzz" of the fly as "blue." This is an example of synesthesia, a literary device whereby one sense is described with an adjective usually used to describe a different sense. In this instance, the adjective "blue" would ordinarily be used to describe a sight, but Dickinson here uses it to describe a sound. Perhaps Dickinson describes the sound of the fly as blue to suggest the light of heaven that she is ascending to, or perhaps the word blue is meant to connote that the speaker is feeling melancholy.
In the final two lines of the fourth stanza, the speaker says that "the Windows failed." These windows are likely referring, metaphorically, to the speaker's eyes. The speaker's eyes "fail," meaning perhaps that they close for one final time, never to open again. This, then, is the moment of the speaker's death. In the final line of the poem, the speaker says that she "could not see to see." This phrase suggests an eternal darkness or, alternatively, an eternal, blinding light. The implication is that the speaker has passed over into the eternal realm of the afterlife.