Corporeal form, when used literally, means the physical existence of something. There's the idea of "you" as a self-aware entity or a soul that defines how you will behave and react, which nobody can see or touch directly; and then there's your corporeal form, which is to say your actual physical body. Similarly, there's the mathematical concept of a square, and then there's an actual square drawn on the sidewalk in chalk. The concept of a square has no corporeal form; the drawing takes the concept and gives it a corporeal representation.
The this that you highlighted is referencing the phrase "quantity changes into quality", and the author then further defines that phrase as being an "allegedly confused, hazy Hegelian notion".
The author is basically saying that you can point to something and say "at the start of this process, we have quantity. Then things happen, and at the end of the process, we have quality. Thus the process embodies the concept of quality changing into quantity."