Quick plot synopsis: Once upon a time, a girl named Chihiro and her parents were driving to the father's new job in a new city when they stopped to explore a strange tunnel. This tunnel lead to the spirit world, where Chihiro's parents were turned into pigs and Chihiro had to work at a bathhouse while she looked for a way to turn her parents back into people. Chihiro met a lot of strange people while she worked at the bathhouse, and learned a lot about herself at the same time.
Yes, the synopsis was written that way on purpose - Spirited Away may be the closest that Hayao Miyazaki has come to telling a European-style "fairy tale" when not adapting someone else's work, although there are more than enough Japanese elements in the story to make it strange to Western eyes.
Other than Chihiro's parents' transformation sequence, there is nothing in this story that a young child would find frightening. However, there are a few things that may start children asking questions that some people aren't ready to answer. Don't use this movie as a substitute for a babysitter; watch it with your children.
Highly recommended.