"We Are Seven" is written in the ballad form, which is an ancient form usually associated with storytelling. The ballad alternates lines of eight syllables with lines of six syllables, and thus has a sing-songy, lilting feel.
In fact back in the day, ballads were meant to be sung. Want to try out singing "We Are Seven"? Pop on over here, play the theme from "Gilligan's Island," and try singing the words of "We Are Seven." (Skip that first stanza though; it's up to some metric funkiness.) If you can sing a poem to the theme of Gilligan's Island, then: congrats. You're singing a ballad. Enjoy the lilting tune.
But that's not all the sound that's going on in this poem. We get a lot of musical sounds in these lines, particularly when the little girl is speaking. Just check out, for example, all the alliteration she uses in stanza 10 and afterward: "graves are green" (37), "side by side" (40), "sing a song" (44), "sun-set, Sir" (45). These lend a kind of bouncy energy to the little girl's character, a sort of buoyant innocence that reinforces her child-like conviction in her attention to the dead siblings. The sounds in her dialogue, then, underscore her character as it's revealed to the speaker (and to us along with him).