What is this part of the mushroom? Is it safe to eat?
This part of the mushroom is perfectly edible. It's a structure called a partial veil which protects the spore-producing gills on the underside of the mushroom cap, usually while the mushroom is still immature. They open as the cap grows larger, exposing the gills and releasing the spores which produce new mushrooms.

It's pretty common to see these on white and brown common mushrooms (which have a huge number of other names) since they are actually an immature form of the "portobello". Since it's a delicate structure, it's a pretty good indicator for how well the mushrooms have been handled and transported; intact veils will usually indicate a recently-harvested mushroom that has been kept well hydrated. If the veil is a little ragged or partially open, that's fine too.

Note that this applies only to the common mushroom; other varieties may or may not have veils.