This is the wrong forum, next time Uncategorized
may be a better fit - but that aside:
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You do not own the rights to rebroadcast the music. That right rests with the copyright owner.
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“Unauthorized distribution or exhibition violates federal laws…” is what is normally stated on media, such as when I look at the back of a Blu-Ray box. Broadcasting on Twitch violates that.
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You could talk to Twitch Legal and see if getting an ASCAP license (http://www.ascap.com/licensing) would allow you to stream music. The license fees are not cheap.
I imagine that Twitch does not have the ability to scan a live stream to check for copyright infringement but has software that does so on the recordings. That is why the recordings are flagged and muted.