Like a lot of old people, Mama Dunsford wants to be as independent as possible. She doesn't want to be seen by others as some sort of charity chase, an object of pity always needing to be helped. People mean well in offering help, but to many senior citizens like Mama, such help acts as an unpleasant reminder of their advanced years, with all the challenges that they entail.
So when Chig reaches out his hand to steady Mama as she gets up, his grandmother shakes him off immediately. It would appear that Chig is laboring under the misapprehension that Mama Dunsford is some kind of decrepit old invalid that can't do anything without assistance.
One can't really blame Chig, of course; not unreasonably, he thinks he's being helpful. Most people in his situation would've reacted the exact same way. Besides, when Chig and his father entered Mama's house, Mama only rose halfway out of her chair, which perhaps indicated to him that the old lady was incapable of standing up on her own. In any case, Mama is anxious to put her grandson right:
When I want help, I'll let you know.
Once Mama stands up, she insists that Aunt Rose can do the cooking, so long as it's not because she thinks that Mama can't. This is yet another expression of Mama's feisty independence, a way of reiterating the point she's already made that she doesn't need anyone's help.