My guess is print on normal paper then glue it to sturdier cardboard (corricaded for example).
Then just build in order:
Background first, them sutable high stack of cardboard pieces (depending how high you want it to pop = how strong 3D effect) , then the printed one on top, then add suitably printed strips of paper to hide the sideview of the plain card board stacks.
BTW: On the page you linked the death mountain area is build very wrong: the pieces should get smaller so there area ledges to walk on, the site shows them getting bigger so they form overhangs with no space to walk on.