You get build up on fans normally due to the boundary effect. Basically, even on fast moving objects, there's this layer of air above the surface where the air is very slow moving (which results in dust settling). By reducing the spacing to the degree they have, they 'cut' into the boundary layer and thus prevent dirt build up.

I'm not exactly sure how transmitting heat across the gap would work. I imagine it would just be by convection, but I don't know how efficient that would be.

That said, I imagine that this type of design would be problematic in laptops. With the spacing so tight, it seems to me that any shocks to the fan could result in the vanes touching the baseplate.