Why does Hester decide to stay in the area? How does her character evolve?

There are a few things to consider here. New England was the scene of Hester's infidelity. Her penance must hence be in the same place. Also Hester's roots, for better or for worse, lies in New England. It was extremely difficult for a single woman with a baby to just pick up and leave. I think Hester is also very proud. If Hester were to leave, she would be letting the powers that be rule her life. She would be acknowledging that her "sin" and beautiful gift that came from it (Pearl) was something to be ashamed of. I think also, she simply didn't want to leave Dimmesdale.