The simplest way to do this is to turn your elements into 3D layers and position them in Z space. Create a Null, make it 3D, parent the other layers to it then keyframe the Null’s Z position. Leave your background as a 2D layer and it will not move with the other elements.
If you want to fade things in/out keyframe their opacity.
You can also rotate the Null and have elements come into view that are above/below/beside/outside your initial view.
A more complex way of doing it is to use a camera – though I don’t think your example uses one.