According to The Word Detective, the etymology of "drag" meaning cross-dressing is interesting: despite a widespread urban myth, it does not derive from an Elizabethan theatrical abbreviation of "Dressed as a girl". Rather, it dates back to the late nineteenth century, although it did originate in the theatre with male actors playing female roles. The use of the term relates to the fact that the long dresses worn in theatrical productions contained a lot of fabric and would commonly run down to the ground and trail along on the floor after the actors, in other words they would "drag". Therefore, roles in which men dressed as women came to be known as "drag" roles. In the Shorter Oxford English Dictionary, the first use in this sense is dated to 1870.
There are, however, suggestions that the term goes back further than this in gay slang. Judy Grahn (Another Mother Tongue: Gay Words, Gay Worlds, Boston Press, 1984, 1990 pp 95-6) also links it to hems dragging on the ground, but says there may be another far older source, based on medieval religious practice. Men dressed as women were sometimes pulled or dragged in ceremonial carts in religious rites in Britain, an event which may or may not have some connection to April Fool's Day celebrations. However, this etymology is rather too speculative.
In usage the word "drag" is typically confined to men dressing as women in show business. "Cross-dressing" is the general term for wearing the clothes of the opposite sex.
References: http://www.word-detective.com/back-p.html
http://www.cuis.edu/ftp/STUMPERS-L/STUMPERS-L.1998-01