The legitimate drama, stage, or theatre, called "legit" in Variety slang, meant straight plays without musical interpolations. The term originated when strict regulations during the 17th and 18th centuries were intended to protect the theatres that were licensed for the performance of plays. Nonlicensed theatre groups would get around the restrictions by adding snatches of song or dance, which theoretically changed the piece into a different (unregulated) genre. By extension, the term "legitimate" came to imply works of some literary merit.
The Historical Dictionary of the American Theater. James Fisher.