I'm a little different from a lot of people in that I get a lot of enjoyment watching people play games. To be honest, I get more enjoyment from watching than I get from participating. I can totally veg out to a random "Let's Play" on youtube, just sitting back and enjoying what's going on.
I think that watching someone else play, either on youtube, in person, or streaming, is relaxing for a couple of reasons. For one, it's like watching a movie that unfolds according to whomever is playing it. Most movies are very predictable in their own ways, and you don't get surprised by a lot. When you're watching someone else play, they bring their own style to the playthrough. There are people who will check out everything, taking their time in each area. There are others who constantly pause and save, over and over again. There are people who rush through all the dialog and cutscenes to get to the action. In that sense, I think you can tell a lot about a person just by how they play games, which is always interesting to me.
I don't play a lot of FPS games (none, actually) but I love to watch others play them. There's always this sense of anticipation from watching others play their game. Watching is the only way I can enjoy an FPS game. I'm not coordinated enough or aware enough to do really well in a game like that anyway. Also, I think FPS games are pretty generic... but when you bring in the idea that every player has a different play style, it becomes interesting to watch. Different people use different strategies, and it's fun to be able to take a step back and analyze gameplay from a different perspective.
I think gaming by yourself can be a little lonely sometimes. When I watch someone else play I feel like I'm participating without actually doing so. I'm generally not the kind of person who likes playing with others online anyways, whether they be friends or strangers. I'm the type of person that plays against bots in TF2 rather than play against other, real people. Call me weird, but watching others play is more relaxing and exposes you to an entirely different perspective.
Anyways, if anyone knows of any good Let's Players or have their own stream, let me know!