Rita is one of the two Marthas assigned to the Commander's household, mainly in charge of cooking. She's strict, disapproves of the narrator, and doesn't want to get involved. It's not as though she dislikes the narrator; she just doesn't want to get to know her. As the narrator says during Rita's first appearance:
Her face might be kindly if she would smile. But the frown isn't personal: it's the red dress she disapproves of, and what it stands for. She thinks I may be catching, like a disease or any form of bad luck. (2.14)
Because Rita doesn't want to know the narrator, the narrator doesn't get to know her. Both Rita and Cora sometimes treat the narrator like one of their chores: more like an object than a person.