Jim,
I'm sorry, I didn't choose the best example in the thread I referenced. My bad. There are better discussions here on Pool Forum.
The important thing to understand is: in a stabilized pool, more chlorine is available where it needs to be to destroy pathogens that enter the water.
Stabilizer binds a percentage (a large percentage but not all) of the Free Chlorine, this protects the chlorine from the sun but also prevents it from sanitizing the water. This applies only to the bound chlorine which is a portion of the Free chlorine. When the portion of the unbound chlorine is used up reducing some "dirt" that gets in the pool, the stabilizer releases bound chlorine to become unbound chlorine and maintain the portion of unbound chlorine to bound chlorine. This newly unbound chlorine is immediately available to attack more "dirt" in the place where the dirt is.
On the other hand, in a pool with no stabilizer, all the Free Chlorine is available to attack the "dirt" but when the supply of chlorine in the immediate vicintity of the "dirt" is exhausted, the remaining "dirt" won't be sanitized until circulation and dispersion bring more chlorine close enough to it.