As its name suggests, a fanlisting (FL) is a non-profit, non-commercial site that lists all fans of a certain subject, primarily for entertainment purposes. An entire community has been built around this concept by and for fans since 2000, so practically every fanlisting on the internet is listed in one of two "master directories": The Fanlistings Network and The Anime Fanlistings Network. (The staff of those sites have been running the networks for almost two decades now!)
Why join FLs? Sites like these are for people who wish to declare how proud they are to be a fan of a subject in the world wide web. Fans don't even need to do anything else after signing up—the whole point is in putting your name down and seeing how many people on the web like the same things as they do.
While not a requirement, a personal website (or fanlisting collective) usually helps fellow fans show off all their badges of interests. In the 2000's, this used to be a pretty addicting hobby.
TL;DR: You're a fan, you're proud to be one, and you want everyone to know. You have nothing to lose by joining. ;)