The story
A jocular reference to a man's tight-fitting swimming costume or swimsuit appearing as if he has a budgerigar concealed inside it, i.e. his bulging genitals.
Etymology from Wiktionary, CC BY-SA 4.0.
budgie smugglers
Today it means: A style of tightfitting men's swimming costume cut like underwear briefs that covers the buttocks and groin but not the legs and reveals the bulge of the genitals; especially worn in surf lifesaving and in swimming races. Full definition →