Timing. I was a Rhapsody user in 2003. I even had an AV receiver that supported Rhapsody back in 2005 or so, assuming it was connected to my computer (for stuff like track listing and whatnot on the device). But it didn't take off because timing wasn't right. People were just at a place to start buying digital tunes -- streaming music was awkward.
But beyond that, it took the iPhone (and devices like it -- but really, let's admit it was the iPhone. I loved my Treo, BlackBerry and Windows Mobile devices too, but it was the damn iPhone) to usher in a device where people could stream on the go. Before that, using Rhapsody (or even Spotify, early on) was kind of a PITA.
Spotify made it easier by letting you sync offline stuff with your iPod or iPhone, but it took widespread 3G and beyond, good data plans and saturation of smartphones for people to be like, "yeah, I can replace my iTunes with streaming music."
The same is, to me, true of streaming video. You could do live streaming close to a decade ago. On your cellphone. Shit, I remember envying people using Qik on the Nokia phones in 2008. But it took a long time, there was massive latency and the quality sucked.
Meerkat and Periscope are fast, easy to use, have restreaming features and are easy to drop-in, drop-out of.
Whether they last or not, I don't know. But just as it took e-sports to make the Justin.tv model work (and I remember when they launched Twitch. All of us, myself included, thought it wouldn't work. But then the devices for livestreaming game videos became so inexpensive and the experience so compelling, Sony and Microsoft built it into consoles. It took the barrier getting super easy and the content being compelling for the user case to be seen).
So again, timing.