1660s, "piece composed in burlesque style, derisive imitation, grotesque parody," earlier as an adjective, "odd, grotesque" (1650s), from French burlesque (16c.), from Italian burlesco "ludicrous," from burla "joke, fun, mockery," possibly ultimately from Late Latin burra "trifle, nonsense," literally "flock of wool" (a word of unknown origin). The more precise adjectival meaning "tending to excite laughter by ludicrous contrast between the subject and the manner of treating it" is attested in English by 1700.
The two great branches of ridicule in writing are comedy and burlesque. The first ridicules persons by drawing them in their proper characters; the other, by drawing them quite unlike themselves. Burlesque is therefore of two kinds; the first represents mean persons in accoutrements of heroes, the other describes great persons acting and speaking like the basest among the people. [Addison, "Spectator," Dec. 15, 1711]
By 1880s it typically meant "travesties on the classics and satires on accepted ideas" and vulgar comic opera. Modern sense of "variety show featuring striptease" is American English, evolved after 1870 and predominant from 1920s, probably from the earlier sense "sketches at the end of minstrel shows" (1857).
A BURLESQUE show, to the average person, is a rather naughty form of entertainment which men attend for the purpose of vicarious thrills and semi-obscenity ["The American Parade," 1927]