In 1950, the musical Gentlemen Prefer Blondes debuted on Broadway and the tradition of the Gypsy Robe was born. A chorus boy named Bill Bradley (presumably no relation to the basketball player-turned-Senator) started it, creating a "lucky robe" from a dressing gown donated by the one of the chorus girls. Since Gentlemen Prefer Blondes was a hit, Bradley wanted to transfer some of that good fortune on to friends in the choruses of other new shows. He passed the robe on to a friend in the next show to open on Broadway, which was Call Me Madam. It was another hit — possibly because of Ethel Merman and a terrific score, but more likely due to that robe.
Since then, with few exceptions, the superstition has persisted for the Broadway openings of shows with a line of "gypsies" (i.e., a singing-dancing chorus). Opening night, everyone gathers on stage an hour before curtain and a robe is presented by a representative of the last show to open. There are all sorts of rules and rituals, and the Gypsy Robe has a glorious history which you can read about here.