This poem is all about singing. Everyone is singing—the deckhands, the washer women, the hatters, and the carpenters. Reading this poem is practically like being at a Broadway play, except instead of singing and dancing, everyone is singing and working. For Whitman, singing is both an expression of the individual and the universal—singing is something we all can do on our own and together as a group. Singing both literally and metaphorically brings our nation together. (Just think of an entire stadium of people singing "God Bless America" during the seventh inning stretch.)
Lines 2-9: Everyone is singing as they work. From mechanics to young seamstresses, each sings his or her own song. They sing "what belongs to him or her and to none else." Their songs are something that come from them, something they own.
Lines 10-11: At night, singing becomes more of a group, party activity. It's something that unites a group of disparate Americans. Americans: we know how to sing a good "strong melodious song."