As the stork delivering two babies is "set upon" by something moving they quickly, the babies (who the stork was trying to deliver to their parents) are separated, Baby Mario hurtles toward the sea while Baby Luigi is taken by the mystery assailant. Baby Mario is fortunate to land on Yoshi's Island with the Yoshi's who discuss what to do with this sky-baby - it is decided that Yoshi must take Baby Mario and reunite him with his brother Baby Luigi but its a long and dangerous road ahead! Check out TMB's playthrough of Super Mario World 2: Yoshi's Island below.
Playthrough / Walkthrough playlist
This playthrough covers all the way from the intro and tutorial world 0 up to the end of world 6, the full ending isn't included in the above playlist, but you can watch it below:-
Super Mario World 2: Yoshi's Island - Ending
Super Mario World 2: Yoshi's Island - All Bosses Video