After Effects - How to create movement across composition

I'd probably make a giant layer, create each animation then parent them to the giant layer (you don't really need a giant layer, but things like CC Repetile can create one for you, but sometimes it's clearer to just use a giant layer) and then once everything is set up, you animate the layer's position and rotation to move different parts into the stage.

Which is pretty much what Oussk said, but if you were to use 3D layers and a camera, you can achieve similar and then add in some cool extras like pulling back, zooming out and maybe even throw in some spotlights.

you can do it in 2 ways, using a camera and a 3D layer or you can make everything parented to a null layer then animate the position rotation and scale properties

There's a dead giveaway on the paper itself: all the edges are identical, so everything is just mirrored. In turn this could simply mean it's not at all a huge composition and just e.g. a CC Kaleida effect with some animated parameters onto which the rest of the animation is overlayed as conventional 2D layers. It still takes time to set up and get everything aligend, of course, but ultimately that may be the simple answer.

Mylenium