what does Nanny mean by “de nigger woman is de mule uh de world?”

In context, Nanny is inferring that although black men are discriminated against because of their race, black women had it worst because as black women, they were the most descriminated against of all.

She is she uses the metaphor of mules and "de nigger woman" to emphasize how black women had to carry of everyone else's weight or burdens—including their own. The rest of the quote explores how eventhough black men face the cruel racist constructs of the south—their pain is ultimately transferred onto their black, female partners or wives. Just like a mule transports everyone's, baggage so does the black woman.