As is often the case with Dickinson's poems, "I dwell in Possibility" features a first-person speaker. This "I," often taken to be Dickinson herself, "dwells" (or lives) in a place of potential—of "Possibility."
"Possibility" itself, in other words, is a kind of home for the speaker—an idea that line 2 makes even clear by metaphorically calling "Possibility" a "House." What's more, this house is "fairer" than that of "Prose"—it's more beautiful, just, and truthful.
The mention of prose helps readers grasp what the speaker means by "possibility" in the first place. Prose refers to regular written or spoken language that isn't poetry. Considering that this is a poem, and that poetry and prose are often (somewhat unfairly!) positioned as polar opposites, it becomes clear here that this is a poem about the art of poetry itself. When the speaker talks about "Possibility," then, she's talking about poetry. To her, poetry is possibility, and this is the world in which she lives.
The alliteration of "Possibility" and "Prose" draws attention to the contrast the speaker is making between these two forms of writing. These lines also have a fairly steady iambic (da-DUM) meter:
I dwell | in Pos- | sibil- | ity –
A fair- | er House | than Prose –
The first line contains four iambs (four da-DUMs), making it a line of iambic tetrameter. The second line has three iambs, making it iambic trimeter. This pattern, known as common measure or common meter, turns up in most of Dickinson's work. The speaker will continue to use this pattern throughout the poem, whose rhythm, sounds, and structure all remind the reader that this piece of writing is exactly what it praises: poetry.