After the coconut hits Higgins on the head, and the Title screen appears, press Right, Left, Right, Left, A, B, A and B. This will bring up the World Select screen. Press Up/Down to select an island, and press A to begin.
You must possess the Hudson Bee from World 1, then hold Left and press Start when your game is over. You can locate the Hudson Bee inside a secret room in the first cave of the first island. Jump up and down after passing the fifth skull to find the entrance to the secret room.