So what the hell does all of that mean?
Short answer: Dungeon World is a Fantasy RPG that's Powered by the Apocalypse.
Slightly longer answer: The main thing that sets games "Powered by the Apocalypse" are structured differently than most RPGs on the market. Instead of having a designated setting set in stone, or maps already drawn, or long backstories or details all planned out, Dungeon World provides some basic resources and lets the players and Dungeon Master (or DM) fill in the gaps.
Did you choose to play The Wizard? Well, there's no one quite like you and your mastery of magic. There can't be another Wizard quite like you - you're The Wizard of the story, and everyone else wishes they were so important.
Did you choose to play The Paladin and set out on a Quest to "Slay Amarakto the Cruel, a great blight on the land"? Well, you and the rest of the people playing the game with you are going to have Amarakto the Cruel as some sort of antagonist.
The whole point is to Draw Maps, But Leave Blanks and Play To Find Out What Happens.
Okay, but why use it for a podcast?
Plus, with it's conversational structure, it works really well recorded as audio.