In LoL, how do championships like LCS and LCK, let alone teams generate money?

LoL actually has their developer (Riot Games) sponsoring the Championship Series primarily with other companies (like Coca Cola, Nvidia, Razer, ect.) sponsoring the tournaments within their regions.  It's highly likely that the various micro-transactions & merchandise help provide the funds (via profits) for the main sponsorship.

In terms of viewship, LoL tournaments tend to pull in around 300,000 viewers over multiple gaming streaming services like Twitch & YouTube Gaming (possibly NicoNico in Japan) in MULTIPLE Languages (so viewership numbers in one stream may be considerably lower as English speaking viewers won't watch on a Spanish or German speaking stream & so on) & a local audience of a few hundred (depending on tournament location).

The tournament series is likely generating enough money to be self-sustaining & the major teams playing in the tournaments likely have consider sponsorships as well in addition of maintaining their respective fanbases.

There's likely some details that I'm overlooking, but since I'm not part of logistics of the Championship Series (being only a small-time stockholder in the developer's parent company), I don't know everything & can only work on publicly obtained knowledge & educated speculation.

I hope this sheds some light on the subject.

They probably get a lot of sponsors.