I believe the reasons they were so successful had to do with four things: 1) They had several band members who could write chart breaking songs. 2) Pretty much all the band members could sing lead which gave the band different sounds, so it wasn't monotonous to listen to them (i.e., New Kid in Town, Hotel California, Life's Been Good to Me). All of these hit songs were sung by different members in the band. 3) Joe Walsh. Not only could he sing and play a variety of instruments (he's mind blowing on electric guitar), he's also a great stage performer, very entertaining to watch. 4) The group's harmonies were almost as rare as The Beach Boys.
After more than 40 years they're still performing, which I find amazing. Here's to The Eagles, one of the greatest bands in our lifetime.