The opening lines of the poem introduce a nameless young woman—"She"—and describe her former fantasy of a bohemian lifestyle.
As the title indicates, this woman and her partner are "living in sin": an old-fashioned way of saying that they're living and sleeping together without being married. (Doing so is considered sinful in some religious traditions.) When they decided to live this "sin[ful]" bohemian life, the woman imagined that their small "studio" apartment "would keep itself." In other words, she didn't give any thought to practical concerns like cleaning the apartment; she figured those things would somehow take care of themselves. She was so romantically smitten that she couldn't imagine any "dust" accumulating on "the furniture of love." Basically, she imagined their love nest as a place where mundane problems either wouldn't occur or wouldn't matter.
To the extent that she did notice or anticipate difficulties, she saw them as part of the romantic adventure of "living in sin." She was so committed to this adventure that she considered it "Half heresy"—almost blasphemy—to wish that the apartment's faucets were less "vocal" (noisy) and that its windows were less "grim[y]." If their love nest were sparkling clean and in perfect working order, she wouldn't find it nearly as romantic!
Notice how, within the context of "living in sin," the religious word "heresy" has an ironic or paradoxical ring to it. Mainstream society may consider this woman's lifestyle sinful, but to her, undertaking it is almost like committing to a faith, and she wants to do it properly.
The poem narrates events in the third person, reporting the woman's thoughts from a seemingly detached perspective. This detachment creates an ironic tone throughout, as reality undermines the woman's fantasy and the reader comes to understand (better than she does) how bleak her situation is.
These opening lines also establish the poem's form: unrhymed iambic pentameter, a.k.a. blank verse. A standard line of iambic pentameter contains 10 syllables that follow an unstressed-stressed rhythm: "da-DUM | da-DUM | da-DUM | da-DUM | da-DUM." In practice, though, the syllable count and rhythm often vary a bit. Look at how this pattern plays out in lines 1-2:
She had thought | the stu- | dio | would keep | itself;
no dust | upon | the fur- | niture | of love.
Line 2 fits the pattern perfectly, but line 1 contains a small variation: an extra unstressed syllable at the beginning. (This isn't a particularly noticeable variation, since "She had" sounds and reads almost like a single syllable, as in the contraction "She'd.")