Right. They don't need all the money. They want to use it for something, so they took lessons from him. Chopin himself also preferred aristocrats as students, rather than normal class people.
Chopin knew the influential Rothschilds through his aristocratic Polish friends, and set up a performance among the Parisian aristocrats who appreciated his talent and invited him to teach. He did not want to teach so many students; so the aristocrats were willing to pay more to have some of his hours.