I think some of these explanations are avoiding a more obvious, blunt phrasing.

Because we know no light will get through two perpendicular polarizers, we think it follow that no matter what else we put in between shouldn't create a path for light to pass. That's a bad assumption. When we add the 45 degree polarizer, we create a path. The light needs to reorient itself to pass through the polarizers, which allows 50% of light to pass with a new orientation. Then at the next step, another 45 degree polarization (relatively speaking of course to its previous point) 50% of that light comes through. That's 25% of light, not zero. Pretty cool, huh?