Because everything would naturally pool to where gravity pulls it, the gasses and liquids the most quickly, the atmosphere and water wouldn't conform to the shape of the cube, but naturally flow to the lowest point. Since gravity pulls everything toward the center of the mass, the liquid and atmosphere would still form a sphere, leaving the impossibly high mountains, or points of the cube, and much of the corners between them, exposed directly to space. This would leave the six faces, unless there was enough erosion on the ridges between them to connect them, isolated from one another, and if the planet can support life, there would be six very separate biodomes that would evolve independently from one another.
What would happen if after gravity naturally did its work, and the ridge between two or more of these separate biodomes finally wore down enough for there to be enough atmosphere for travelers to go from one face of the cube to another? (As it appears it already did on the left side of the picture.) Mind you, while it looks flat from space, to someone on the surface of the planet, because of the way gravity works, the further a person would get away from the middle of the face, the more it would feel as if he or she were climbing an increasingly steep hill. Even with serious erosion, going from one face to another would be a tough job. But it would make a pretty cool story! Here's an interesting article that explains the idea of square planets better than I can: http://www.askamathematician.com/2011/05/q-what-would-earth-be-like-to-us-if-it-were-a-cube-instead-of-spherical-is-this-even-possible/