In nearly every neighborhood there's the Gladys Kravitz, the snoop, who sees all and knows all. In the small town of this poem, that busybody is our very own speaker. Every detail we get about the death in the house across the way is through the eyes and ears of this speaker.
And just in case you fell for the illusion that poet and speaker are one and the same, gotcha. This speaker announces, in line 12, that he used to wonder about the details of death when he was a boy, making him at the time of this poem a man. So, no, our speaker is not Emily Dickinson herself.
A silent and mostly objective observer, this speaker acts as a kind of sentry, watching the traffic in and out of the dead one's house and remarking the hour and its passage. The only shades of sentiment we get from this cool customer is that he thinks the minister is acting bossy, "As if the House were His — And He owned all the Mourners — now —" (14-15) and the undertaker's job is appalling, "the Man / Of the Appalling Trade — (17-18). You get the feeling that what's most galling is that there is a flurry of life and trade and work surrounding an eternal stillness.
The calm plainness of the speaker's language is oddly reassuring, don't you think? The point of view keeps the facts of this death at a safe distance. The only exception is when the mattress is tossed out and the kids scurry from it, thinking the worst. Most of the facts of this death are seen from far enough away (or imagined behind closed doors) that there isn't such an impact. We readers can be just as calm, cool, and collected as our speaker.