Actually, not all of them do. The practice has its origins in the dictates of modesty. From time immemorial, an uncovered head was considered immodest. Married Jewish women covered their heads, usually with a scarf or veil, so as not to draw attention to themselves.
In relatively recent times, the wearing of a wig or a half-wig, called a “sheitel” in Yiddish, became a way for a very religious Jewish woman to conform to the requirement to cover her hair. Eventually some decided that it was easier (and cooler) to wear the sheitel on a shaved head or one with short hair than to cover a full head of hair with a wig. It also has the advantage of not allowing one’s real hair to show, thus preserving modesty. However, some ultra-orthodox believe that wearing a wig looks too much like real hair and so will only wear a scarf or veil with no wig.