Collect the all oil barrels in the indicated level.

Then press X to have Funky fly you to the corresponding bonus level.

Little Chill 'n' Char Island: The five oil barrels in Chill 'n' Char Island.
Little Ghost Island: The five oil barrels on Ghost Island.
Little High High Island: Both oil barrels in Little High High Island.
Little Lost Island: The five oil barrels on Lost Island.
Little Sun Sun Island: The four oil barrels on Sun Sun Island.