I had the fortune to try and tutor my girlfriend (now wife) in Physics when we were undergrads. Learned very quickly that using big words, formulas, and assumptions is a very bad way to illustrate a point. Sorry if I insult your intelligence by simplifying too much.
Heat is energy. Temperature is often called the "average kinetic energy" of a substance, but I don't find that definition very intuitive. I prefer to think of it as "heat density" which is not a very good use of physics terms, but is accurate enough. Temperature is how much heat there is in a certain amount of matter. (The problem is that the scales don't actually relate to a heat density but are more a relative scale...enough said, doesn't help understanding).
If you compress a portion of matter into a smaller volume, you are also putting more heat energy into that volume and are increasing the "heat density" or temperature.
So the quote "Since heat does not leave the affected air mass, this is an adiabatic change of pressure, with an associated change of temperature." means "Since you aren't changing how much heat there is, but are compressing all that air into a smaller space, the temperature will rise."
Matthew.