A hairdresser is someone who cuts and styles hair. To braid hair is to fold three sections of hair over each other repeatedly to create a design. "Braid away" sounds close to "fade away". "Fade away" means gradually become weak, especially to the point of death. The joke is centered around the term "braid away" since it sounds like "fade away" and the joke is explaining what happens to old hairdressers.
OLD HAIRDRESSERS never die , they just braid away . .