Fan-what?
A fanlisting (FL) is a non-profit, non-commercial online listing of all fans of a certain subject, mostly for entertainment purposes. In this case, this site is an FL for the Dairy Queen Blizzard.
Why bother with a fanlisting? For website owners, a fanlisting serves the same function as any other listing -- more exposure for your site, since it gives you more backlinks, possibly improving your pagerank. For non-website owners, it's a fun way to see your interests immortalized online. :p