My friend has an old Victorian house with original quarry tiles in his
kitchen. The tiles are about half an inch thick and he asked me if
there was an easy way to remove stains (mainly fat I think), then
re-seal them.

The best chemicals are available from Lithofin. www.extensive.co.uk. They
have a range of cleaners, up to severe aggression, although as others have
stated, they might not be entirely effective if the fat is now internal to
the stone. However, once you have managed to clean them, they do a range of
excellent sealers. I'd probably go for KF StainStop for quarry tiles.

Christian.