Not surprisingly, a linen closet is a closet for linens: sheets, towels, etc.
In a modern American home, the linen closet is often near the main bathroom. It's smaller -- less deep and wide than many other closets and most of the storage space has shelves rather than hanging space or cubbies.
Older linen closets in wealthy homes used to have places to hang linens to dry or air. They were sometimes larger -- walk-in size -- with bureaus or chests for storage.