How do professional League of Legends players make money?

The way most professional e-sports players make money off games is by live streaming, YouTube, brand sponsors, and/or tournaments.

Live streaming: Pros will download streaming software such as OBS or xsplit to stream the game they play (such as League). Sometimes you will solo stream or stream with your team. Once you gain lots of fans/watchers, they apply for a partnership with the service. This will allow them to get money for ads/subscribers. This is usually done through twitch or MLG.com (MLG is rumored to give you more money per ad view)

Twitch partnership: http://www.twitch.tv/p/apply

MLG partnership: http://www.majorleaguegaming.com/news/become-an-mlg-tv-streaming-partner

Youtube: making videos. Whether it be highlights from your latest 5k mmr match (or whatever league uses) or full-length gameplays with commentary, uploading videos to YouTube is another way to earn money. (Same as above,) you have to gain enough subscribers then apply for some partnership. While I wont give any link as there are many, many partners, you can search that on YouTube itself.

brand Sponsors: These are for more after you've done the other three, but it's worth mentioning. Brands are starting to sponsor esports quickly. AT&T and monster energy even sponsored the dota 2 team evil geniuses. Your team may get some once you come in a high place in the first tournament.

Tournaments: once you team becomes pretty skilled, you may be invited to a tournament somewhere. Use this to capitalize on the possibility of sponsors like said above. These are the top earning methods that give pro players their money.

I did start rushing this at the end, sorry if it loses quality over time.

To add upon what @Judge2020 said:

Players playing in the LCS (as well as in any other region) earns a "split" salary. Meaning that they receive this amount each split. This salary is given to the team as a whole and then distributed by the team owner/manager. The minimum amount that needs to be payed towards players are ($12,500/€10,000).

This amount is per player.

2.2 Player Compensation

Each team must distribute the required Minimum Player Compensation ($12,500/€10,000 per starting player per split during the 2015 season) to its starting players, in accordance with the terms of the applicable Team Agreement.

Source: 2015_LCS_Rule_Set_2.01

Expanding on the brand Sponsorship answer, there are also esport agencies that get behind elite players and work in very similar ways to traditional sports agencies to make the player (and agency) the most money. One esports agency even has a full competitive gaming training facility in LA that trains the most elite players by keeping them on strict diets and exercises specifically made for pro eSports gamers. It's nuts, but exciting at the same time. Basically, there are lots of ways to make money in eSports.