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.
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.