How can mail get cross-contaminated with anthrax?

Only by design. Ordinary mail including packages will not pick up things like anthrax or other infectious agents. For this to happen, someone would have to, on purpose, open one piece of mail and insert anthrax. There is no chance for accidental contamination.
Spores. If a person mails anthrax spores by mail, the spores are small enough to penetrate paper and contaminate other pieces of mail that come in contact with the piece containing the spores.
Exposure. If one letter in a bin or pouch contains harmful bacteria, all the surrounding mail would also be exposed to the bacteria and become contaminated just as a rotten apple can spoil the whole bin.