I'm also curious about that. I thought every royal family was funded by tax payers.
A lot of it has to do with the arrangements they've made with the government. The UK family as well as the Civil Service and the Military used to be funded by the Crown Estate, but George III made an arrangement that the government would get the lion's share (public costs) in return for an allowance for his own expenses. Technically the British taxpayer doesn't really fund the official business of the Royal Family --the Crown Estate/Civil List revenues do. Yes, yes, there are still some costs borne by the government (like security), but it's peanuts.
The Monaco situation is different. It's a tax-haven, so no income tax to its citizens, but the head of state is a part-owner of the casino operating company. So he makes bank.