What does supersaturation with respect to ice mean?
The saturation vapour pressure defined by the American Meteorological Society:

The vapor pressure of a system, at a given temperature, for which the vapor of a substance is in equilibrium with a plane surface of that substance's pure liquid or solid phase; that is, the vapor pressure of a system that has attained saturation but not supersaturation.
The saturation vapor pressure of any pure substance, with respect to a specified parent phase, is an intrinsic property of that substance, and is a function of temperature alone. For water vapor, the saturation pressure over supercooled liquid differs appreciably from that over ice.

When water vapour exists over a body of supercooled liquid water, water is constantly exchanged between the liquid form and the water vapour form.  When this reaches equilibrium, there is equal phase change in both directions.  If the air becomes saturated, no more water vapour can be added to it, for it will condensate to the liquid form immediately.  At this temperature, the vapour pressure is the saturation vapour pressure with respect to water.  Remember that water can exist well below 0°C — clouds are typically not completely frozen until temperatures drop to around -40°C, although it depends on many factors and the details are subject to ongoing research.
When water vapour exists over a body of ice, there is a phase exchange.  At a certain temperature and pressure, the air is saturated, any water added will precipitate as ice.  At this temperature, the vapour pressure is the saturation vapour pressure with respect to ice.
By &quot;Body of&quot; 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.