In "Before the End of Summer" by Grant Moss, what do you think the grandmother means when she says to Dr. Frazier, "He'll have to go to the Stewarts'"?

She means that Bennie isn't able to live on his own yet; he's only ten and can't live without proper guidance. His grandmother is the one who takes care of him. Since she's going to die, he's going to need oversight and guidance from a different person to make sure that he grows up happy and healthy.
Grannie is trying to make plans to ensure that everyone in her family is taken care of before she passes away. For example, she wants to make sure that her daughter marries Joe so that she will have a good life. She tries to take care of all the problems herself, even suffering through the knowledge of her impending death alone. However, before she dies, she finds out that Bennie knows the truth and has been working to make her life easier.
Miss May, Grannie's friend, was going to take care of Bennie. However, she died before Grannie and that meant that Grannie needed to make alternate plans. She's a woman who believes that she can help guide her family through life even after she's gone. Part of this includes making sure arrangements are made for her family.

Bennie's grandmother has been informed by the doctor that she doesn't have long to live; she'll most probably pass on before the summer is over. This doesn't give her a lot of time to get her affairs in order. But Grannie's a very strong, practical woman and immediately starts making plans. One of the things she has to take care of before she dies is what will happen to Bennie. The young boy is only ten years old; as Bennie's grandmother is his sole guardian, it's important to find someone who will take on the responsibility for taking care of him when she passes away. The old lady suggests that Bennie will have to go and live with the Stewarts. She clearly feels that they will be able to take care of him.