Watching live streams of people playing video games is a massive and impressive phenomena and at some point will dominate the media landscape and games industry in the future!
This might sound to you like a very controversial standpoint, so let's take a step back and look at what is happening right now.
The most prominent representative and market leader of streaming services is twitch.tv formerly known as justin.tv. Twitch is owned by Amazon (bought in 2014 for $970 million). With the Stream First Program they present some ideas about game design for live streaming.
Another player is Youtube, market leader in online video, but not as strong in streaming as twitch. Youtube is owned by Google (bought in 2006 for $1.65 billion). In 2015 they introduced youtube gaming, a video-gaming oriented sub-site and app to compete with twitch.
Then there is Mixer formerly known as Beam, which was a small new startup until it was aquired 2016 by Microsoft for an undisclosed amount. It was rebranded and is now deeply integrated into Xbox services. They also have a lot of interesting features for 'crowd play' and innovative viewer interaction and promise less delay than other platforms.
Of course Facebook has its own live streaming service Facebook live and Twitter aquired the live video streaming app Periscope for $86.6 million in 2015. Afaik they don't have specific game-related features, though.
So you see, most big players of the media industry are already heavily invested. Let's concentrate on live streaming games, because IMHO this is currently the most advanced form of streaming. Especially when you look at monetisation methods.
Why is it so popular, you ask? I want to play myself and not watch someone else!? Streamers are not just playing a game, they produce a show. Maybe they are exceptional good in the game they are playing or they are funny and charismatic. There are many reasons to watch other people do stuff. This is not any different from TV.
Even more important: They produce a live show with viewer interaction. People interact with the streamer and other viewers, they start to greet each other and build a community. This is new and very different from a live TV show or archived videos. The sense of community can be very strong and this is used (or even exploited).
It all started with viewers donating money to the streamers, basically paying them for their services on a voluntary basis.
You probably know that you can subscribe to channels on twitch paying currently $4.99 per month, which gives you a badge visible in chat and the ability to use special emotes in chat. (FYI: twitch keeps half of the money)
But did you know, that you can buy "Bits", a digital currency on twitch, which you can spend for cheering in chat with a special emote? Additionally you can now buy (selected) games the streamer is currently playing directly from twitch and the streamer gets a 5% revenue share.
Oh, I forgot to mention "Drops" and other "loyalty programs" to encourage viewers to keep watching and be an active member of the community... This is a totally new and experimental Eco-system right there.
Will it some day blow up, be replaced by something else or just grow until we all watch streams all day? I don't dare to guess what the future will bring in this highly dynamic industry, but currently it seems like live streaming is here to stay, constantly growing and evolving.
All economic aspects aside, this phenomena is still super interesting from a media perspective:
Games, movies, shows and events are and will be used as content for streaming outside of their intended form, adding a layer of entertainment brought to the viewers by the streamer in his role as player, moderator, host and entertainer. This does not replace other forms of entertainment but is built on top of it.
Exploring new ideas of viewer interaction to create games (or movies or shows) tailored to this format may lead to interesting novel and unique experiences in this new form of entertainment. There are already some small examples like the game "Choice Chamber" or the Liveshow "Chat Duell" by RocketBeans (basically family feud with viewer integration), but in my opinion there is still a lot of untapped potential.
This wont be the last article on that topic here as I'm currently researching in this direction and working on a game for streaming with deep integration of viewer interaction.
Live streaming (games) is massive, but viewer interaction is still in its infancy. It is a very interesting new combination of media and should be researched more... Which I am going to do!
 Twitch Stream First Program
 Mixer Introduction
 Research Paper "Streaming on Twitch: Fostering Participatory Communities of Play within Live Mixed Media"
 Game "Choice Chamber"
 "Chat Duell" on RocketBeans.tv