I would imagine that the creators of that game created a level editor, which exports information about the various rectangles/spheres (I didn't see any other primitive), which the game then understands to turn it into basic openGL (or other API) calls.
From what I understand, a pretty common workflow (especially for 3D platformer games) is to create a level definition using basic geometric primitives to define the playable space (and physics interactions, etc.).
Once that is in place, it basically becomes a skeleton into which all the rendered objects are placed.
@snape Consider this: a simple room can be drawn in OpenGL using six quads. Each quad is defined by four points. Re-using points, you can define a room by eight points.
You can make a new texture from a png or other image, set a scale for the texture, then only apply the texture to a mesh. As far as I know you can't even align the texture
@thedaian There are at least a handful of free plugins that let you export to .X
I guess I'm a bit biased since I can actually use Blender, so I'd have that level set up pretty quickly, but Blender (and most 3D programs) has this ridiculous learning curve at the beginning
@snape For a Friday deadline, I would probably draw out a level by hand (using only quads). Define the vertices, and for each quad, enter the coordinates into a text file.
@JohnMcDonald: the reason I even brought it up is because it seems like a really easy shader to write. it's basically just a small script that looks like "if close to a integer value of x,y,or z return white else return black"
@snape: whether you should use blender kind of depends on are you currently capable of loading and displaying model files in whatever code you have so far
well the absolutely simplest thing to do is to try it out in immediate mode. draw a cube with a texture. the texture being a glTexture from an image of a grid