By "Body of" you should probably not so much think of a frozen lake or liquid ocean but rather of small cloud droplets or ice particles.

This graph from Vermont State College shows the saturation vapour pressure over water or ice:

Source: Vermont State College

As your source stated, this difference is important for meteorology as it affects the formation of clouds and precipitation.

Now as to why the saturation vapour pressure is different with respect to water than with respect to ice is another question. You may find more expertise for that on Chemistry Stack Exchange than here on Earth Science. My atmospheric science textbooks in any case don't go into detail there.

I made a graph (below), similar to the phase diagram above, but does not clip off the clausius clapeyeron curve. Notice that below 273 K, the saturation pressure of vapor w.r.t. (with respect to) ice is lower than water. This basically says that it will first turn into ice before it turns into water. Then, above 273 K, the vapor saturation curve wrt water is less than ice- meaning it is easier to turn into liquid water above 273 K than it is for ice.

Next is the concept of supersaturation. Supersaturation generically refers to a state that is above saturation. In this sense, supersaturation can be identified by having a relative humidity over 100%. While some websites may give a different definition, I think of relative humidity as being the ratio of water entering the liquid (or solid phase), to water becoming a gas. So supersaturation means that there is enough water vapor in the air, at that temperature, to form a cloud.

I should note, that the process of making a cloud can be a bit more convoluted than this answer. For that, I recommend looking at this answer where I discuss Kohler theory.

So the short answer is, there is sufficient water vapor in the air to form a cloud. But because it is below freezing, the cloud is formed via deposition, much like how frost is formed.