What is Ibsen’s use of his set design, and how does it help the story?

Ibsen is very specific about his description of the set, which, according to his stage directions, is meant to evoke “a room furnished comfortably and tastefully, but not extravagantly.” Part of Ibsen’s intent is to establish the social standing of his characters, and to invite the audience to draw some inferences about their lives from the details of the set. To this end, he includes directives about the placement of a piano and “engravings on the walls; a cabinet with china and other small objects; a small book-case with well-bound books.” These details suggest that the inhabitants of the house, though not very wealthy, are educated and enjoy an above-average income.
Of course, another gloss on the stage description is the title of the play itself. In being so specific with his stage directions, Ibsen is of course making another “doll’s house” out of the stage; he is creating a series of minature spaces in which characters can interact, and perhaps also making a comment on the middle class trappings of the set, which can be understood as a furnishings for a doll’s house, or, in other words, as playthings.

The question asks about Ibsen's use of the set design, and I am interpreting this part of the question as Ibsen's purpose for offering the set directives at the start of Act I of the play. I hope that works for you.
The set directives are thorough, which reflects the clarity with which Ibsen imagines the situation in which he places his characters. He specifies a piano, a round table, arm chairs, a small sofa, a stove, two easy chairs, a rocking chair, a small table, a cabinet of china, a case of books, and engravings on the walls. These household items all help the audience understand that this story is one that concerns a sociable lot, as the numerous places to sit suggest a lot of merriment and conversation. The china also contributes to this impression, as it reveals a dignified host who seeks to respect their guests by serving refreshments in china. Additionally, the books and engravings communicate a sense of culture without overwhelming the guests or the audience with showiness.
Ibsen's set directives help establish the tastes and the interests of the Helmers, providing the audience with an impression of the family and their home that does not take up any of the dialogue.