Emily Dickinson's poems can be enigmatic and hard to understand. In poem 591, "I heard a fly buzz - when I died," she is describing the impossible, namely, her own death. Her spirit is speaking to the reader after her body has expired, and she is describing what it felt like to go through the process of dying. Stanza two describes the mourners, stanza three describes making her last will and testament and reintroduces the fly, and the final stanza continues to describe the fly and then her final demise. Here is one way to paraphrase the first stanza, emphasizing its literal meaning: "When I passed away from this life, I heard a fly buzz. There was no other sound in the room. It was kind of like when a thunderstorm comes through, then it gets calm, and then another thunderstorm comes through. That's how quiet the air felt felt, and that's probably why I could hear the fly so clearly."
Here's another way to paraphrase it, reading a little more emotion into it from the rest of the poem: "I remember well the moment I passed from my previous life into this one--the day my spirit left my body. One of the weirdest things was that a fly buzzed just as my spirit left. It was a fitting sound effect, perhaps, to designate how paltry and insignificant one's physical existence is in the whole scheme of things. Yes, people had been crying up a storm when they knew I was about to pass. At that particular moment, though, no one was crying. I guess they were all cried out. But soon, when they realized I was gone, the wails would resume. But at that specific moment, it was as still as the calm that occurs between two successive thunderstorms."