Why are shrouds so called?

Standing rigging (typically the metal cables nowadays, but basically a line that doesn't move) is named based on it's use and orientation relative to the ship. Stays typically run "fore and aft" or parallel with the ship, and these can be used to keep the masts in place, and to hold up sails such as a jib or a staysail. Shrouds however run perpendicular to the ship and hold the masts in place on that direction. As for why they're called "shrouds" specifically, quite frankly I don't know.

As a side note, the shrouds are only the vertically running part, the footholds that make it possible to go aloft are called "ratlines" which run horizontally.