I don't know about stains and taking them away, you could try some white vinegar and/or baking soda to clean it. I do know terracotta tiles (especially unglazed ones) should be cleaned and rubbed in with soft soap (the most oldfashioned type you can get). There are lots of fancy products available, but terracotta tiles have been used countless years longer than there were chemical cleaners and waxes.

Personally, I would call a company that specializes in this sort of thing, like Coit or possibly Stanley Steamer. They're probably your best bet and will also guide you on the proper sealer so this doesn't happen again.

Sand off the sealer, wash the tiles with a non-acidic cleaner, when dry reseal.