Is there a specific reason why this ship is vertical? Or just because we can?
The boat basically becomes it's own anchor, anchoring itself in the calm water under the surface.
Think of it this way: The vessel in perfectly calm water will sit at a particular level. Now a wave comes, and briefly the force acting on the ship is "proportional" to the buoyancy volume, ie height of the wave and the area of water that the hull intersects. Nothing can be done about the wave height, but the waterplane area can be reduced, minimising the force. The price you pay for that is a low payload.
Now if the vessel had a natural frequency of bobbing up and down close to the frequency of the waves, a resonance could occur which would amplify the motions. But the natural frequency of such a structure is by design very low, many times lower than the wave frequency, so resonance does not occur.
Other structures which operate on the same principle are semi-submersible drilling/production vessels (see http://en.wikipedia.org/wiki/Semi-submersible) and "spar" type oil production platforms (http://en.wikipedia.org/wiki/Spar_(platform)). The need to support a heavy payload on these structures makes the design quite challenging, as a good compromise must be found between hull cost, payload and low motions.
Interestingly, another type of oil platform, the tension-leg platform, takes the opposite approach of having a very stiff anchoring system and a high natural frequency to achieve the same effect of low vertical motion in heavy seas.
(I am an engineer in the oil and gas industry with many years of deepwater development experience).