As the story opens, the grandmother is reading a newspaper article about the Misfit, a criminal who has broken out of prison and is heading for Florida. Florida is where she and her family are driving for a vacation. The grandmother announces forcefully she wouldn't take her children in the same direction as the Misfit, as it isn't safe.
We find out in the first sentences of the story that the grandmother wants to go to Tennessee, not Florida. Bringing up the Misfit being "aloose" is an attempt to persuade Bailey at the last minute to change his mind about their vacation destination.
Talking about the Misfit characterizes the grandmother as manipulative. Being powerless, the grandmother will use whatever means are at her disposal to try to get her way. This opening, in which the grandmother's words are quoted directly as a dialogue, also reveals that for all her pretensions of being a lady, she does not use the most polished speech. Instead, she uses words like "aloose" and terms like "look here." She also stands over Bailey with her hand on her hip, "rattling the newspaper at his bald head." This is not a pose normally associated with a refined lady.
The introduction of the Misfit at the start of the story also foreshadows the encounter he will have later with the grandmother and her family. Ironically, while the grandmother is only trying to use him to get her way and is ignored by her son, she is correct that the Misfit poses a grave danger to the family.