The dark world was a central part of the 1992 SNES original--it was the realm where longtime series antagonist Ganondorf had been imprisoned--and required hero Link to have the Moon Pearl in his inventory; otherwise, he would transform into a pink rabbit upon entering.
A common motif of Zelda games is being able to travel to alternate worlds, with Ocarina of Time allowing Link to travel to a dark future and Twilight Princess featuring its own twilight world.
Also in the interview, Aonuma said that the game was being designed specifically for the 3DS' lenticular screen and that players wouldn't be able to experience A Link to the Past 2 without having the console's 3D slider cranked to maximum.
Aonuma also says that the idea of Link being able to transform into a painting originated from the Phantom Ganon boss fight from the N64 classic The Legend of Zelda: Ocarina of Time. Phantom Ganon attacked Link by hiding in six identical paintings.
The Legend of Zelda: A Link to the Past 2 will be released for the 3DS later this year.