The speaker, Whitman, is a poet celebrating the various sounds, occupations, and activities of the American people, and he uses the phrase "the varied carols" to depict this diversity. The metaphorical usage of "carols" conveys the unique expressions of each individual, likened to song.
Through out the poem, Whitman describes the sounds and actions of carpenters carpenters, mechanics, farmers, mothers, and more, each singing a distinct tune. This inclusive approach implies that every single American, no matter their background, occupation, or origin, contributes to the broader identity of the nation.
The speaker, Whitman, hears these diverse voices and perspectives, and through them, he captures the essence of American identity. Thus, the line serves to emphasize the pluralistic nature of the nation, its multitude of voices, and its shared experience.