one of my friend (mentor really) is something of a judo (and martial art in general) historian. has done lots of research on judo in the US. Basically, what I have learned from him is that JA was once the Armed Forces Judo yudanshakai (in the JF). They split in late 60s. (He gots theories about that, but I'll stick to what I can remember as fact) AAU was already "in power" because of the way sports organization in the US was run at the time (they ran all the amateur sports). Congress passed some act in the late 70s which established the USOC. What came from that was that each sport had to have 1 governing group that reported to the USOC, which led to the formation of US Judo Inc (now USA Judo).
why are there still 3? basically, because the powers in each have different ideas/methods on how things should be done (again, just trying to stick to facts, not paint anny swaying opinions), but today, they are working together more than they were way back when they split (apparently, there was a lot of disagreement. so much to the point that years later, they don't know why they don't like the other guy, but just that they're supposed to).
Which should you join? I asked this very same question. The (surprising) response that came back made great sense. All 3 have their pluses and minuses, but in the end, the job of the organization is to serve the membership, so whatever organization is providing the most service in your area is almost always the best bet (why do you want to join an organization when your the orphan member in your area? (yes, you might have a good reason, but I did say "almost" always)). The caveat there is that if you start competing nationally, most tournaments require membership in USA Judo, so if you're with one of hte others, you might have to pay for a second membership.