Topologically, it's a disk with two holes and an outside edge, but it's clear there's trilateral symmetry. Here's the

This piece corresponds to the grey heptagons:

Now we join two of these pieces together to get half of the Klein quartic:

And here's its skeleton:

There are four heptagons on each piece that interlock with each other (where grey meets green).

10,20,16,15

12,5,13,24:

Now we add two more pieces to get all the heptagons (though we haven't connected all the edges yet):

The corresponding skeleton--

--curls up into a tetrahedron:

It's possible to make the paper model curl up like that, but it gets so crumpled that it's not worth taking a picture of. It should, however, illuminate the structure of Klein's quartic enough that you can match up the heptagons in this hyperbolic tiling to the ones made of paper: