if authors get an advance that's greater than the sale of their books, doesn't it mean that the publishers lose money, i.e. paid more for those books than the books sales?

Authors get around 10% of the cover price of a book as royalties, it depends on several factors. The rest goes to the publisher. So some do lose money, but the break even for the publisher is usually well before the advance is fully covered by royalties.

Well the publisher also pays for the book to be bound, edited, overhead for their staff, cover art. Many books don't sell for the full retail price and are discounted. So net of all of this a 10% profit margin is common, they aren't keeping 90% of the book sales.