In a nutshell, Twitch is a video platform similar to YouTube that's primarily geared to streaming people playing video games.
Players, sometimes referred to as streamers, will play video games and narrate or commentate on what they're doing, why they're making that particular decision or move in a video game - all of which can be watched by anyone. In some instances, it's just gameplay - as in, you only see what the player is actually doing on screen, with their commentary running over the top.
The more common form is a PiP (Picture-in-Picture) where one camera will be on the player and then another on the gameplay itself. This is why you often see some Twitch videos with a small representation of the player in the bottom left corner. They're playing with a green-screen in the background that's then removed to sit more seamlessly with the gameplay footage.
Viewers go to Twitch, and can search by either the game they want to see streams of, or particular streamers to see what they're playing. In the instance of searching by a particular game, viewers can sometimes be looking for tutorials or playthroughs, which for platform games and RPGs, can be useful as they're full of tips and tricks to clear a level. In the case of streamers, it's more for personality and the sense of fandom that they engender.
Like YouTube streamers, Twitch streamers have a lot of earning potential. Just this week, Richard Tyler Blevins - better known by his Twitch handle, Ninja - faced questions over reportedly accepting a $1 million payment from EA Games to help promote 'Apex Legends', a new free-to-play game being touted by the publisher. Ninja is the most-followed video game player in the world, and has a total of 13 million followers on Twitch.
Technically speaking, there's nothing illegal in what he did. Video game publishers will regularly pay streamers - if they have enough of an audience - to play their games in order to help promote them. More than that, most streamers have to disclose that they've been paid to play certain games. Ninja, in particular, was most likely chosen to help promote 'Apex Legends' as he's known primarily for playing and streaming 'Fortnite', which is a direct competitor for 'Apex Legends'.
As of May 2018, Twitch usually has 1 million concurrent viewers. That means at any moment in time, there's at least 1 million people either streaming or watching someone's stream. There are an estimated 2.2 million broadcasts per month, with a further 15 million daily visitors to the site. Like YouTube, Twitch has a partner program where popular streamers earn a cut from any advertising revenue that Twitch makes on their stream.
Not only that, some Twitch streamers have private channels where viewers must subscribe in order to watch, the cost of which is divided between Twitch and the streamer themselves.
As to why anybody would want to watch someone else playing video games instead of playing it themselves, the best analogy is to look at sports. Sure, you can go outside and kick a ball around a field, but watching someone at the top of their field playing it something entirely different. Most Twitch streamers spend upwards of 20 hours a week playing a particular game, and very often play at a professional level, i.e. esports tournaments. Not only that, some streamers are given first-access to new games that have huge fandoms, meaning that those subscribed to their channel will get to see gameplay footage before anyone else.
Since its launch in 2011, it's become a success story and forced YouTube into offering more livestreaming options for players. Twitch, meanwhile, has moved into more unconventional areas for streaming. Twitch Creative, which is aimed to anyone in a creative field such as painting or dance, launched itself by streaming episodes of Bob Ross' 'The Joy Of Painting' non-stop for eight days.