A standard innerspring mattress is light for its size but floppy and awkward. Bagging it and carrying it flat, rather than bent, keeps it clean and protects the internal springs.
Pro tip: A mattress on its edge acts like a big sail; on a windy day two people should keep a firm grip crossing open ground.
Watch out: Do not fold or bend a standard innerspring mattress; it can permanently damage the coils and ruin the support.