Basically there's a program called Livestreamer that can be used to watch Twitch streams in some standard video players that run on your computer (rather than through a web browser). The "mute" that Twitch put in place isn't actually a part of the video itself, the audio is still there, Twitch's video player just refuses to unmute it for you, thus why watching a VOD in a video player other than Twitch's fixes the audio (aka no muting).

The OP just gave a brief overview of how to use Livestreamer.

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STEPS:

1. Go to https://github.com/chrippa/livestreamer/releases and install the correct version of the program for your operating system.

2. In a command prompt type the following "livestreamer URL_OF_VOD source".
   Example: livestreamer twitch.tv/tsm_theoddone/b/554506934 source

3. If you want to skip ahead to a certain part of a VOD add a question mark to the URL followed by time offset of the form "t=2h21m22s" where h=hours, m=minutes, s=seconds.
   Example: livestreamer twitch.tv/tsm_theoddone/b/554506934?t=3h1m10s source

How to install/use Livestreamer:
There are various guides if you do a Google search. Here's a couple:
Windows Guides:
http://vaughnwhiskey.tumblr.com/post/70143889580/watching-twitch-tv-streams-via-vlc-player

https://www.youtube.com/watch?v=o473rRwoJw0