In "Winter Dreams," how does the little girl make Dexter quit?

The spoiled little girl in question is Judy Jones some years before she grows up and captures Dexter's heart for the rest of his life. At the age of fourteen, Dexter meets Judy on the exclusive Sherry Island golf course really for the first time, although he had seen her several times the year before. Even as a young girl, however, Judy is beautiful and willful, full of the haughtiness of a child of wealth and privilege. When asked to caddy, Dexter explains that he cannot since he has been told to stay until the caddy master arrives. When Judy cannot have her way to play immediately, she pouts. She is not used to being told "no." 

Dexter cannot stop watching as Judy then vents her temper upon her nurse who has accompanied her to play golf. When Judy tries to strike the hapless woman with her little golf club, Dexter unexpectedly identifies with Judy rather than the Jones' family employee:

He could not resist the monstrous conviction that the little girl was justified in beating the nurse.

When the caddy master arrives, Judy just assumes Dexter now will be her caddy, but he does not move to pick up her golf bag. The caddy master speaks to Dexter, his hired help, without respect, accusing him of "standing there like a dummy" and telling him to "Go pick up the young lady's clubs."

Without thought or hesitation, Dexter quits, giving up the job he needed very much. His snap decision frightened him, but he did not change his mind:

. . . he had received a strong emotional shock, and his perturbation required a violent and immediate outlet.

It is not so simple as that, either. As so frequently would be the case in the future, Dexter was unconsciously dictated to by his winter dreams.

Dexter quits his job because he rejects his role as "hired help" for Judy Jones and the glamorous social class she represents. He chooses not to carry anyone's golf clubs, either literally or figuratively. Even at the age of fourteen, when Dexter's romantic dreams of a wealthy and glamorous life clash with his middle-class reality, he chooses his dreams.