I understand the mathematics behind it but I am looking for an intuitive description.
Higher amplitude waves, on water, are caused by larger disturbances and longer durations of energy input, in the form of wind, than smaller ones. Quite simply in order to build a big ocean wave you need to put in a lot of acceleration of surface waters over an extended period of time. So it's not just that the wave itself is travelling that much faster, the whole surface that the wave is moving on is itself moving in the same direction as the wave on the surface. The larger amplitude of the waves is in fact partly due to them going faster rather than the other way around.
NB: This doesn't apply with Tsunami waves as those are full column disruptions involving the water column from seabed to wave crest.
This falls under the basic principles of physics - the higher the amplitude, the more energy. The ocean is the material that is being used, but think of it as an isolated wave of energy. Under any application - light, sound, etc - the higher the amplitude a/o frequency, the more energy. More energy = more speed.