The blackness on the sides isn't a huge deal - it's pretty standard if you're not running at 16:9, and people with the same monitor size can maximize the window and it will properly fill up their whole screen.
This is more of an issue with the way twitch designed the web page; not particularly your fault or anyone's fault. Your monitor is simply not 16:9. There are only these things you can do:
1.) Set base resolution to 1920x1080 and fill in the black borders with a custom image/background to maintain aspect ratio (probably the best thing you can do)
2.) Set base resolution to 1920x1080 and "skew" your game's image with the game capture options or by stretching the source if it's not game capture (Note: stretching/skewing your image is not recommended, I hate stretched streams personally)
3.) Get a standard 16:9 monitor
4.) Leave as is, the blackness on the sides don't bother me at all personally
