Like with any published expanded universe stuff, reading about stories in between or beyond what's portrayed in the official media is what I like about fanfiction - be it exploring the established world building and universe more or specific interactions and/or stories between characters.

When I was waiting for books in a series to release (typically there's a 1-2 year wait) I began reading fanfiction, especially if the book ended on a cliffhanger.

A desire to do better by a couple of side characters who I felt got mistreated by their series. Admittedly, I don't read fanfics at all but I write two.

i didn’t like the ending of a book series i read way back in middle school and i went online to search for “(book series) better ending” never found a really good one for that series but i’m glad it got me into fanfic

my love of Harry Potter + Kingdom Hearts!! Back in the golden days of the early 2000s

I've discovered fanfic itself around the mid-2000s. I was neither a particularly active reader, nor was I bound to any canonical material.

What really got me going was when I discovered the Chip 'n Dale Rescue Rangers fandom in late 2005. I had been a fan of the show since its German premiere in early 1991, and I was glad there was a fandom full of like-minded people. And especially, there was fanfic.

I had read some before I joined the fandom and the main forum. Mere hours later, I think, someone suggested I could start writing fanfic. Many of the few fans wrote fanfic in those days. First the suggestion seemed weird, also since I'm not a native English speaker. But then I decided to give it a try.

Let's just say that I was bursting with ideas just days later. That's how my fanfic-writing craze of 2006/2007 began. A pity it slowed down a lot afterwards.

By the way, I refused to read the often-recommended classics at first. I didn't want them to influence my writing too obviously. I guess they did later on, or my ideas for certain details were too similar.