OK, Simon Says,
If you were asleep that day during your Private Pilot ground school, here's how CG affects stability. Assume a conventional airplane: one wing, horizontal stabilizer or stabilator aft of the wing, standard empennage, T-tail, or Cruciform tail. Stability of delta wings, canards, flying wings and other types are outsde of the scope of this explanation, but unless you're doing your Private Checkride in a Piaggio or an F-102, I doubt you'll be asked to explain gc and stability for one.
First, you have to understand why an airplane like this is longitudinally stable. The cg is ahead of the center of lift of the wing. Because of this, the horizontal stabilizer must provide a downward force to keep the airplane from pitching forward. What happens to your airspeed when you pitch up and start climbing? It decreases right? The downward force of the horizontal stabilizer (like any airfoil) is a function of airspeed. Pitch up, less airspeed, less downward force, aircraft pitches down. The change in airspeed changes the downward force of the horizontal stabilizer, causing it to be stable. The same effect causes the airplane to respond to downward pitching by pitching up. Pitch down, more airspeed, more down force on the horizontal stabilizer, aircraft pitches up
Now, up to this point, we have assumed the cg is where it is supposed to be. The question is: Why does an aft gc cause an airplane to be less stable? Well, let's look at the extreme example of aft cg, where the cg is actually aft of the center of lift of the wing. In this case, the force of gravity acting behind the wing will cause the airplane it pitch up, so the horizontal stabilizer must provide a force UP to keep the plane from pitching up. You make it create a force up by trimming the angle of incidence of the stablizer (if it's trimmable) or by trimming the position of the elevator or stabilator. With the gc this far aft, what happens when the plane pitches up? When the airspeed decreases the UP force on the tail decreases, this allows the plane to pitch up more, more pitch up, less airspeed, less up force, more pitch up, and it just builds on itself. Without any control inputs from you pretty soon you're pointing at the sky, wishing you hadn't loaded those bricks in the tail. In engineering terms, this is what is known as a "positive feedback loop" The opposite happens in response to a pitch down, more airspeed, more up force on the tail, tail goes up, nose goes down, airspeed builds more, and if you don't do something, pretty soon you're pointing at the ground, watching your wings come off.
This is what happens when the cg is well aft of the cg limit. The airplane doesn't go instantly from "stable"to "unstable" as the cg is moved aft of some magic point, rather the stability gradually decreases as the cg is moved aft, until you reach a point where you have a plane that is a handful to keep level. Even within the acceptable cg range, you can sense the difference in stability.
Now, we could go a lot deeper, examining how changing angle of attack on each of the airfoils adds to or decreases stability, How the location of center of pressure on each airfoil moves with aoa, changing each airfoil's contribution to pitching moments, thus affecting stability, etc. I'm sure I could write pages, and not come close to covering it all. However, the increasing/decreasing airspeed explanation is fundamentally correct and substantially complete. It can be found in most good private pilot texts, and as such, seems reasonable fodder for a checkride question at any level.
regards