In short, the two "screens" are now side-by-side on the single screen. The room you're currently in displays on the left, with the next room on the right; at any time protagonist Alta can "flip" to the next room, trying not to be crushed by the new surroundings.
Mighty Flip Champs DX will be released on the PlayStation Store July 12. In the announcement after the break, designer Matt Bozon encourages new players not to "miss out on this top shelf experience. Even if it sounds weird."