To get past obstacles, you have to warp between screens. You can then, for example, walk across the bridge where there is one, and then warp back to the other screen, where you're now safely across the gap. A "ghost" image of your character appears to help you properly orient yourself in the other world.
That's the basis for some downright evil situations, like having to jump up on a series of floating, vertically stacked platforms that happen to be on opposite screens, or segments of a ladder distributed the same way. You'll have to quickly dodge in and out of the world to defeat strategically placed enemies, and make timed jumps onto moving platforms on different screens.
It gets crazier. Future levels introduce a gravity shift, in which the different screens are oriented upside down from one another, so you'll flip upside down when you flip screens. This, of course, is used in jumping puzzles in which you have to flip a couple of times mid-air to float under a platform and land on the other side. Other levels have winds on one screen you must avoid and/or use to your advantage (for example: jumping, then screen-flipping into a strong horizontal wind to extend the distance of the jump), or dangerous heat that limits your safe exposure to one screen.
Then there are shmup levels, in which you're actively encouraged to flip screens to shoot enemies on both -- despite the temptation of using the mechanic to avoid bullets or oncoming ships. You have to constantly destroy enemies to fill meters on both screen, and if one runs out you fail.
Speaking of failing, I spent a lot of time doing just that. I was profoundly embarrassed to be performing so poorly in front of Endgame Studios co-founder Grant Davies, but that's probably the best thing I could have done. This is a game that requires more than reflexes. You have to adapt to new, strange circumstances, and it doesn't hurt to spend some time memorizing the layout of each stage.
You might be wondering how the 3DS's stereoscopic display works -- why one world would be 3D and the other wouldn't. Endgame deals with this by making neither world 3D. Both screens display images made up of polygons, but the other kind of 3D isn't present. Having to switch focus rapidly and repeatedly between 2D and 3D would probably lead to a different kind of stress than that produced by the game design, anyway.