Why do married women cover their hair?
The reason that married women have to cover their hair essentially is because it is ossur m’dorayso for her t go outside with her hair uncovered. (See gemorah Kesubos 72a and Brachos 24a) It is learned from the fact that the torah says that a sotah’s hair would be specifically uncovered when her husband brought her to the Bais Hamikdash. From here we see that a married woman’s hair must be covered. Regarding the reason that a married woman’s hair, and not a single girls hair must be covered, is for a number of reasons. See Livosha Shel Torah 21, that he brings a number of reasons for this. One reason is that since now she is an eishes ish, and forbidden to marry anyone, therefore she should dress in a more tznius way, so that other men should not be interested in her, and as a statement that she is already married. Another reason is that essentially all women should cover their hair; however a single girl was not obligated in this aspect in order for her to get married.