The poem's opening lines set the scene: it's the morning after someone has died, and there's a "bustle," or flurry of energy, "in a House." Readers can assume that the deceased's loved ones are rushing around as they deal with the practical considerations that arise in the aftermath of a death.
The speaker doesn't go into detail regarding these chores and duties, but they might include making arrangements for the funeral, contacting other friends and family, cleaning the house in anticipation of visitors, and so on. In short, death can create a lot of work—a lot of hustle and "bustle."
These lines set up the poem's meter. For the most part, the poem uses iambic trimeter. This refers to lines of three poetic feet, each of which follows an unstressed-stressed syllable pattern:
The Bus- | tle in | a House
The Morn- | ing af- | ter Death
This steady rhythm perhaps conveys the swift efficiency with which the people in this house go about their death-related chores. The enjambment between these lines (and, indeed, the entire stanza) adds to the sense of hurry. This flurry of activity, of course, contrasts with the lifelessness of the deceased.
Note that there's also a pun in line 2 on the word "Morning." While this refers to the actual time of day, it's also a play on the word "mourning"—that is, a period of grief. This is both a morning full of immediate practical concerns and the early stages of the grieving process. There are two realities going on at once, running in parallel: the mundane tasks made necessary by the death, which keep everyone occupied, and the thornier emotional labor going on below the surface.
That "Bustle," then, can perhaps also be taken as a metaphor for a busy internal world of thoughts and feelings in the wake of death (an idea that the second stanza will explore further).