Not like youtube where it's free and it means you're "following" them, here you pay money and 50% of it goes to the streamer. The viewer gets some emotes usually for it. Twitch has a following system as well, but just different verbiage lol.
You can 'gift' subs and it will just pick some people randomly (that are watching the stream) and subscribe those selected people to the streamer, like if you and I were watching Yoshida stream and you decided to gift subs - you would pay the value of the sub and then maybe I am lucky and get to claim that as if I subscribed (in the value of emotes and such). There may be other benefits to being subscribed, like if they have a streamer store (common for warframe). Where you watch the streamer and then get to spend points (based on time watched usually), to buy items that the streamer will trade to you in game (where being subscribed might earn you double points). Colors, icons, special chat features, etc may be involved.
There are tiers as well so you can spend more to earn more emotes, and maybe other benefits (streamer by streamer case). These emotes also show up in your discord if you connect them, so you can use joeshmoe's silly face emoji in discords. If you're a streamer you get your own emoji automatically (cause you can't sub to yourself), so.. if you were realaaaaaallllyyy determined to have custom emotes but didn't want to pay anything.. just become a twitch partner and then upload emoji you want to use on discord. Lol. A lot of effort. Though I don't stream anymore I still have a few emoji, 'for freeeee' lol.
You can also donate, depending on how, that % to streamer can be much higher than a subscription - like if someone has a paypal. Related to that is bits, another twitch pushed system, that works similarly to donating to paypal (but twitch makes a bit off the top). Like bits though a subscription is promoted by twitch via a number of ways (because they make money off that themselves).