Because 90% of all music enjoyed by most people is under copyright and searching for music not under copyright that you can listen to and genuinely enjoy is not anybody's idea of a good time.
Nobody wants to worry about this stuff.
As long as the purpose of the stream is not just streaming music i don't see the problem however I'm certain the MPAA and RIAA would not agree.
Bear in mind, copyrighted music doesn't just mean "Stuff from the top 40s". Music that's in the soundtrack of a game is under copyright.
So the answer to the question "Why do you have to use copyrighted background music for streaming of a game" is "Because playing a game with the volume at zero kind of sucks".
edit: And obviously that's not the entirety of the concern; there are definitely some streamers who play essentially radio music (or anime music or whatever) over the soundtrack on broadcast. I agree that stuff like that doesn't necessarily need to happen. But using a broad brush to say "No broadcasting copyright music on twitchtube" is both not a good idea, and exactly what people are worried will happen.
To be honest, you're living on borrowed time. The only reason twitch would allow this is because it has flown under the radar so far. It's not a reasonable expectation that you can forever have a service that will let you stream copyrighted material to people.
If they are good or not, the laws exist. Once Twich is under Google's umbrella, they will be pressure by the content owners to enforce those copyrights. Maybe that's not happening today because Twich's community is smallish (compared to Youtube) but once they are "integrated" into the copyright notification/enforcement framework, end of story.