How does the organization of the poem shape what the poem says?

Let's start with a simple overview of what the poem says. The narrator is dying, people have gathered around, and the entire room is a peaceful place to be. Unfortunately, a fly enters the room and interrupts the peaceful dying process.
The poem itself is organized and structured to highlight this peaceful atmosphere that gets annoyingly interrupted. The poem is composed of four stanzas of four lines each that are very tightly organized. The lines are written in a perfect iambic rhythm. This means that the syllables alternate from unstressed to stressed throughout each line. I'll use the first line of the second stanza to illustrate. The stressed syllables will be in bold.

The Eyes / around /- had wrung / them dry -

Additionally, lines 1 and 3 of each stanza are tetrameter, and lines 2 and 4 of each stanza are trimeter. This very regular rhythm and meter gives the poem a very singsong feel to it. It's peaceful and it flows well. It perfectly matches a romanticized idea of somebody peacefully going to their death with loved ones nearby.
But remember, the peaceful death is interrupted by an annoying fly. Dickinson's format mirrors this by placing a bunch of dashes in the poem. These dashes force the reader to pause in inconvenient locations, and the flow of the verse is thrown off just like the smooth and peaceful death is inconveniently interrupted by a buzzing fly.