The APR rules appear to be fairly detailed. I haven't found the actual rules, but the
APR Data Collection Guide has a lot of information.
Here's a brief and rough description works. As usual there are exceptions and special cases.
APR data is collected each semester. The first issue is which players are in the cohort, i.e., who will be included in the APR calculation. Only players who are counters, and who have been at the school for the first 5 weeks of the term are included. The results for a particular term aren't known until after the term is over. Players get 1 point for being academically eligible for the next term, and 1 point for retention, defined as still being enrolled after 5 weeks into the next term. For a few players, only one point is possible. As an example, a graduating player can get a point for academic eligibility, but retention isn't meaningful. Players can be described as scoring 0/2, 1/2, 2/2, 0/1, or 1/1. The APR is calculated as the total of the points of every player in the cohort, divided by the total points possible. That gives a number like 0.959, and that is mutiplied by 1000 to get 959 as the score.
Statisticians will appreciate that the NCAA has put a lot of work into finding a method to convert scores from quarter system schools to match semester system scores (the quarter system gives 1.5 times as many opportunities during a single year to get retention points), and to deal with the statistical validity of small squad sizes (one bad apple has a much bigger effect on the golf team than the football team).
Anyway, what is the effect of cutting players? Let's take the UGA case, where one player was cut (or left) before the fall 2008 term, and 6 other were evidently cut after fall 2008 practices. Any of those players who did not have athletic money (weren't counters) in spring 2008 weren't in the cohort, and have no effect on the APR. Those who had athletic money last spring but not durng fall 2008, and stayed at school (whether or not they were on the roster) through 5 weeks, have no effect. Any cut players that stay at UGA with good grades through 5 weeks into the spring 2009 term will score 2/2 for fall 2008, and won't be in the spring 2009 cohort.
In other words, players fitting the description above will either have no effect or a positive effect on the APR.
Players who had athletic money in spring 2008 and left school before 5 weeks of the fall will lose 1 point for retention. Similarly, players who had money during fall 2008, and who leave before 5 weeks into the spring 2009 term will loose a retention point. They would also lose an academic eligibility point if they didn't finish the fall 2008 term in good standing.
The considerations are the same for cut, drafted or withdrawn players. So if a team loses a lot of players to the draft, it probably can not afford to cut other players because they would be inclined to transfer.
If a team loses fewer players to the draft than expected, and those players are at risk academically, the value of players who are academically strong goes up. It might make sense to "un-invite" incoming freshmen, who haven't shown yet the ability to handle college classes, and keep the existing players. That may have been a consideration for UGA.
Coaches have to plan on players losing retention points through the draft, or by transferring. So coaches should be glad to find an incoming player who isn't at risk of losing eligibility points.