The significance of the scene in which Mama breaks the dishes is that it shows that she is a proud woman who is not willing to be taken advantage of. She is determined to maintain her sense of dignity and to, as much as possible, avoid being used by the people who are abusing her and her family.
Mama breaks the dishes when the government has given the family a mere 48 hours to move farther away from a particular naval installation. She cannot take the dishes with her because there is no room in the car. A second-hand dealer offers her only $15 for dishes that are worth $200. Instead of letting him cheat her like that, she destroys the dishes.
By doing this, Mama is showing that she has her pride and her dignity. She would rather get no money while keeping her pride than allow this man to take advantage of her.