The reason for this was that Kiki would sometimes get lost in thought and start to lose altitude without noticing. The bells went up after she had a head-on collision with a telephone pole, totalling her broom and bruising herself. It was the only wish of Ms. Kadono, the author of the original book, that these bells be rung when Kiki leaves her hometown. Because that's the requirement. A witch girl has to leave her home at 13, and has to live a year in a town with no witch to establish herself as a full witch.
Witches are now rare and weak, compared to ancient times.
This custom of leaving home at 13 was established to make sure that witches wouldn't become extinct. There are many theories. The radio may have been picking up short-wave programs. But one by one, she loses these. Only after that does she gain her true independence.