If you look at condensation fog through a strong magnifier, you'll notice that the fog is actually composed of a large number of hemispherical water droplets. The optical effect is caused by the fact that the tiny droplets act like lenses, scattering the light. On a vertical glass panel like a mirror, the maximum stable droplet radius is strongly affected by the surface tension of the liquid; with a high surface tension, large hemispherical droplets are stable.
If you coat the mirror with soap, however, the surface tension is vastly reduced, and thus there is a reduction in the maximum stable condensation droplet radius. As a result, instead of condensing as a large number of hemispherical droplets which scatter light, you get a uniform thin film of soap water, which has no optical scattering effect.
So it's not so much that you're eliminating condensation (the water will still condense), but rather you're changing the geometric (and thus optical) properties of the water droplets that condense on the surface.
Admittedly, this is a guess, so if anyone has any corrections feel free to point them out.