A magic carpet, also called a flying carpet, is a carpet that can be used to transport persons who are on it instantaneously or quickly to their destination
It originates from one of the stories in One Thousand and One Nights in which Prince Husain, the eldest son of Sultan of the Indies, travels to Bisnagar (Vijayanagara) in India and buys a magic carpet
Carpet is also slang for pubic hair. There’s therefore a slight indication that Aladdin is offering to perform cunnilingus—a “magic carpet ride”—on his to-be lover; and, consequently, the rest of the song takes on a distinctly sexual flavor: “a whole new world,” “dazzling place I never knew,” “indescribable feeling,” “I’m like a shooting star…” etc. and so on.
