Are you sure you're right about that ummmm.........? I've never heard of a Birmingham Southern in I-A, only UAB...
Okay, technically, since I-A and I-AA are purely football designations as Matt pointed out above, Birmingham-Southern was Division I, not I-A or I-AA. Since the only college sport I watch is football, I just said I-A out of habit. But yes, they initiated a move from NAIA to Division I in 1999 and then a move from I to III in 2006.
As to why I-A teams play I-AA teams, NKVD, it's a function of scheduling and economics.
Every I-A team except Hawaii is permitted twelve regular season games (Hawaii is permitted thirteen basically because they need an extra game to offset their god-awful travel expenses). For conference-affiliated schools -- that is, everyone except Notre Dame, Army, Navy and Western Kentucky -- the majority of those games are against conference opponents, half at home and half away, on average.
So that leaves most schools with around three or four non-conference games to try to make money on.
College football is separated into fairly well defined 'have' and 'have-not' groups.
Teams in the have group want as many home games as possible, since they regularly sell out and reap huge financial benefits for the athletic department (and surrounding community).
Teams from the have-not group typically do not sell out their (much smaller) stadiums and do not receive as much of a financial benefit from playing at home.
So if you're a 'have' school, like Michigan, you're faced with two options when it comes to filling your schedule.
First, you can play another 'have' school. The trouble with that is the other 'have' school will most likely demand a return visit. This is known as a 'home and home' series, where one year a school will come play on your field and the next year (or sometimes in a few years) you will go and play on their field. So you get a big financial benefit in the year they come to your place -- since it's a 'big' game and will attract more attention than a game with a 'have not' school, perhaps even including additional television revenue -- but you lose a home game -- and the attendant revenue -- in the follow-up year when you must visit that team.
The other option is to play a 'have not' school. A 'have not' will typically visit a 'have' for a guaranteed payday ($400,000 for Appalachian State at Michigan, as mentioned above). This works for both schools since the payout is less than the money the 'have' school would lose playing a home and home series with another have school, but more than the 'have not' school would make by playing at home. So it's a win-win for both schools, financially speaking. Also -- again, as mentioned above -- playing a 'have not' school gives a 'have' school a break from their more rigorous conference schedule.
Okay, once you've got all that figured out you've got to a) find a good 'have not' school to come to your place and b) have matching off weeks.
A 'good' 'have not' school means someone close to you geographically (lower travel expenses means lower payout) but not so close that it will hurt your recruiting if you don't beat them as you should. For example, Alabama typically plays 'have not' schools from Louisiana and Tennessee, states that are very close to Alabama but that they do not typically recruit heavily.
Finding such a team with a matching off week is practically an art form. That's why you'll often see a 'have' school playing the same 'have not' year after year; it's just too hard to find someone different to play.
So, short answer: Nobody
wants to play a I-AA school, it's just that sometimes a I-AA school is simply all that's left.