Well, there's a few things. Note, I'm not an expert on any of this stuff, just someone who managed to succeed in a beginner's level outing. (Which is probably more helpful to you.) For better advice, you should hit up Edmon or someone else. (I wonder what systems the HBS folk were using for running their game during their streams!)
Streaming a game really relies a lot on getting a good solid connection for one half of it... and basically having a machine which can reasonably off-load some of its power to the encoding process. OBS is the program I use, and I stream to Twich, because both are free and simple to get started with. (Of course I could always sit down with someone experienced and tweak the settings to eke out the best performance but that's another thing.) I had a laptop which was donated graciously to me - an ASUS Republic of Gamers G75VW model - which when given a wifi adapter and allowed to start up fresh? Worked rather well for about 2-3 hours before the workload would require me to give it a break, as the encoding started dropping frames rapidly.
The homebrew system I made is based around a Ryzen 5 and an older video card (Radeon R9 280 3 GB TurboDuo) and using PC Part Picker to fill in what parts would work. (Though in the process I learned a bit about it, my brother is making the next one to be a bit more flashy and higher-powered.) It has streamed for 6 hours and not really done much - but I need to be careful the rest of the family isn't streaming something on the Playstation 4, or other stuff, because it affects my connection just a tad.
