The real genius of bittorrent is how it brings together many people who are interested in sharing the same file all at once so you get great download/upload speeds. This mostly isn't even due to the protocol itself but in the way that torrent sites tend to present the torrents chronologically.
Plus there is no bandwidth wasted on sharing content other than what you are interested in. Almost every other client wants to make you share all kinds of other stuff to even use the network: emule, freenet, tor, i2p, gnutella, etc.
Finally, pure luck has allowed the laws to progress in a way that hasn't resulted in the immediate shutdown or blocking of popular torrent sites. It should be noted that in certain countries this isn't the case and bittorrent has been a total failure there.
Also, on the majority of American college campuses today, bittorrent is a huge failure because it is easily fingerprinted and will get you banned from the network. Maybe they have finally added some stenography to protect against this but not that I know of. Then, there are the large ISPs that are degrading or blocking bittorrent now.
Of course, everyone knows that the real reason bittorrent beat freenet is because bittorrent was python and freenet was a big steaming pile of java:)