The opening stanza introduces the poem's setting: a graveyard. It also introduces the poem's two characters, both of whom are dead. One is the speaker, who "died for Beauty"; the other is a man buried next to her, who "died for Truth."
The speaker reports that they'd hardly been buried themselves—"was scarce / Adjusted in the Tomb"—when the man "was lain" to rest in an adjacent grave, or, metaphorically, "an adjoining Room."
From the start, then, the poem establishes an eerie, Gothic atmosphere (pretty common in Emily Dickinson's poetry!). Though its scenario is supernatural, the speaker presents it matter-of-factly. The speaker doesn't explain why or how the two of them died for beauty and truth, but these claims suggest that they're both martyred idealists of some kind.
"I died for Beauty" hints that the speaker is an artistic or romantic type: someone who suffered for their art, suffered in love, or both. Dying "for Truth," meanwhile, might suggest a political or spiritual martyr: someone who stood up for their convictions and was killed for their honesty. Or it might refer to a scholar type who worked themselves to death in pursuit of knowledge. Then again, these two could have had the same temperament and/or occupation, since lines 5-8 go on to suggest that "Beauty" and "Truth" are the same thing!
It's impossible to know for sure what happened to these two, and that's part of the point. The closing lines of the poem will cast their idealistic sacrifice in a coldly ironic light, suggesting that it loses importance once they're in the grave. Their life experiences, and the very ideals they died for, soon fade from memory.
The first stanza establishes that the poem is a ballad, like most of Dickinson's poems; that is, it's a rhymed quatrain in common meter. Common meter refers to lines that alternate between iambic tetrameter (four iambs, or da-DUMs, in a row) and trimeter (three iambs) with an ABCB rhyme scheme. For example:
I died | for Beau- | ty—but | was scarce
Adjus- | ted in | the Tomb
Ballad stanzas are often used in narrative poetry, including poems about tragic love and death. The ballad stanza is also used in many Christian hymns, like the ones Dickinson grew up hearing; Dickinson often used this form to question traditional Christian ideas about death and the afterlife. (Again, as she does here!)
The stanza contains two of Dickinson's signature dashes: the one that marks the caesura in line 1 ("I died for Beauty—but was scarce") and the one that ends the stanza. Dickinson frequently used dashes in place of commas and periods, and she sometimes inserted them in places that wouldn't normally call for a grammatical pause. This effect gives her poems a slightly tense, staccato quality, which adds to their dramatic effect.