It means that being an American is a source of pride and that is why people sing with their mouths open, that is, they sing loudly so that everyone can hear how proud they are.
Explanation:
Although you did not show in the question above, the line you present is part of the poem "I Hear America Singing" where we are introduced to the American working class and how they do their jobs happily while helping to build the country they love so much. The poem shows how important these workers are and reinforces how proud they are of their roles, as well as being extremely proud to be American, even going so far as to sing about it "with open mouths," which means that they sing loud enough so that everyone can hear well.
