These brownish "age spots" are an aesthetic problem, but not damaging
foxing. Most sources say it's called "foxing" because the color resembles that of a fox's coat. However I found one blog which said it comes from one of the common sources of foxing, Ferrous Oxide, or rust.
The older a book, the more likely it is to experience foxing. However, newer books which are exposed to the wrong circumstances (like high humidity and inconsistent temperatures) can suffer from foxing, too.
When examining a book, it is important to distinguish between foxing and mold or mildew stains. Foxing is essentially a cosmetic issue, while mold and mildew can both disintegrate paper and cause health hazards as well. This gets particularly tricky with older books, which may smell musty simply because of their age.
Mold and mildew stains are typically either dark (sometimes purplish, black, or dark slate gray) or white. I have a huge problem with white mildew growing on the outer fabric covers of hardbound books, because I live in an area which frequently experiences high humidity and cool temperatures, and I tend to keep my house cool.
Foxing on the other hand is tan to reddish brown, and is often faint (at least in the earliest stages). There are two possible causes of foxing, but even experts are undecided as to which is the most common culprit:
Biological: many of the stains seem to be made by a particular group of fungi. Studies have shown these fungi are found on foxed pages, but not on unfoxed pages. Furthermore, when cultured in a lab, these fungi "create new browning stains when reinoculated into paper under laboratory conditions."
Chemical: paper - particularly old paper - has a lot of stuff in it. Microscopic metal flakes can end up in paper from the paper-making machinery. Tiny flakes of iron and copper can oxidize and create stains on paper.
The best way to prevent foxing is to keep your books under good conditions. Provide air circulation (both in the room and behind the books - never push your books all the way back against the back of the book shelf), steady temperatures between 60 and 75 degrees, and a constant humidity between 30-40%.
Removing stains from books is a task best left to expert conservators. Unfortunately, any cure for foxing is likely to destroy the paper more than the foxing will. Luckily, the age spots of foxing are not destructive. Consider it just a bit of added patina.