Earlier quoted context omitted.
Depends on the game. I have watched Dota 2 games and have absolutely no idea what anyone is doing or why. I tried playing HOTS and even the tutorial couldn't explain to me what a MOBA is. I think a lot of Twitch is not competitive games like Dota or Overwatch and are really casual games that anyone can understand. I never played Portal 2 cooperative but can easily understand what's going on if someone is streaming it…
This is why I've always thought Rocket League would take off in a big way, it's as trivial to understand as most physical sports. So far I've been proven wrong, but who knows, maybe it'll turn around.
Overwatch I got sucked into immediately. I knew the basic idea; kill the red team... and the rest of the details I picked up as I played. The matchmaking did a very good job of keeping the games winnable but not boring. The variety of heroes has something for everyone; from those that have been playing FPSes since they were 5 (Soldier, McCree, Widowmaker), those that picked up Overwatch as their first competitive FPS at 30 (Winston, Reinhardt, Moira, Mercy...), or those that want something completely different (Doomfist, Sombra, Symmetra). I have made many friends in that game, even met my (ex-) girlfriend via "stay as team" after a good match.
The biggest problem Overwatch has is that they made it appealing to the hard-core gamers as well, so you have to wade through all the people in the community that complain about "no aim, no brain" because they happened to play Aim Hero 13 hours a day and believe that's the only skill in a first-person perspective game that involves killing people. And the balance of the game is such that right now, they're kind of right... people with average aim in other games can absolutely destroy with Hanzo given the large hitbox of his one-shot-kill arrows. You don't need positioning, you don't need teamwork; just get those arrows in the immediate vicinity of the enemy's heads and you win, no matter what your team is doing. They'll probably tone him down a bit in the near future, though.