For me, writing fanfiction was a three step process.
First, I wrote some original fiction, or started on it at least. It all started with a dragon and a bazooka in... oh hell, first or second grade. From then to about fourth grade, I added chapters to that story. Around the time that I finished the fourth chapter of that admittedly silly piece, I started into reading the sci-fi/fantasy genre heavily. I think I was in fourth or fifth grade when I started reading the Apprentice Adept novels, which anyone who has read them will tell you are fairly advanced and a bit mature. In seventh grade, I read Virtual Mode, by the same author, and... well, I won't go into details, but it changed my life. My entire perception of reality shifted at that time, and I was pretty depressed. I was in a pretty bad funk until about ninth grade... when I really started writing seriously. It's also around that time that I got a computer and achieved access to the internet, go figure.
I had been interested in D&D for a while, but had no friends to play it with and so didn't go any farther with it. But then I got to the internet, and found out about online Role Playing, and decided to try my hand at it. After a short-lived e-mail-based RP by the name of Legends of Quithra that consisted of three people, including me and the GM, I took the simple character I had created there and the base elements of the storyline and built an epic tale... in my head, at least. It was a fairly pathetic story, but my entire ninth-grade English class was shocked when the teacher asked me to stop reading and tell her how many pages I had, receiving an answer of twelve... and I had told her that it was barely begun before I started... serves her right for telling us to write a three page 'epic' :no:
Shortly after that, during my softmore year in high school, I met a young crack-artist (who had broken the protections on all the school computers and installed Unreal Tournament) who was, at the time, obsessed with the webcomic Sluggy Freelance. For the next three months, every morning on my mother's fast-interneted work computer, for about an hour, I would read that comic. After that, I began reading more webcomics, eventually ending up with a collection of about thirty... which I eventually lost contact with due to sheer volume. Sluggy's the only one I read anymore, mostly due to what happened shortly after... I decided to join the forum and discuss some new comic that was particularly shocking, and in doing so discovered the Role Playing sections of the board. After a bit of scanning, I joined a fledgling RP... Adventurer's Union, a silly RP with silly characters, tossing in everything from Miroku to Laharl at random, and before it's untimely death, there was mention of the Lake of 1000 Drowned Pandas. I entered my first RP character, Scygnus Darkhawk. You guessed it, Mary Sue Self-Insert From Hell. I'm kind of embarrassed to mention it now, he was pretty bad. But from there I entered more and more RPs, interacting with all sorts of people and creating a myriad of weird characters... and then a talented young lady by the handle of Kiroth6 introduced me to proper, real anime. Before that, DBZ and Sailor Moon dubs were all I had seen. She introduced me to Evangelion and Inu Yasha, and after the horrible dubbing job on some of the minor villains of the second series, convinced me that subbed is the way to go. Between her, the bunny, Himi, and Krail, I ended up with a considerable collection of anime knowledge.
From there, it was only a matter of time until I discovered fanfiction, though I quite forget exactly what I read first or how I found it... I think it might have been Kiroth, or maybe someone else, that directed me to the first piece. I think the first place that I frequented was a Slayers Lina/Zelgadiss shrine, which resulted in my first fanfiction, which is still on my FF.net account that I would be ashamed to link anyone to until I get something at least decent on it. I found FF.net shortly after, and then MM.org. The All Powerful Hawk's awesome fanfiction lead me here, and the rest, as we say, is history.
I write... because it makes me feel good. I count my characters as friends, ones that won't leave or betray me. In return, I make sure they have a happy ending... though getting there isn't always so fun.
As for why I write fanfiction, it's the reasons stated above, mostly. It's easy, and simple. Several of my originals were created within existing worlds, such as Raifu, who hails from The Land of Stephen R. Donaldson's 'Chronicles of Thomas Covenant' series, and is a powerful Magic Knight of Cephiro besides. Some were created for RPs in existing worlds, like Theon of Tenmei Higure, a Megaman based RP from the bunny, who I then proceeded to tie in with Raifu using the Half-Made Blade and the two halves of Satan's Face, which tied them in with the completely original(at least, drawn from the normal earth) but omnipresent Max Crowley, current ruler of a moderately sized Hell.
The fanfiction I write with established characters is usually written to address what I feel is a need to fill a certain gap that fanfiction has not decently covered yet.
Pretty much, This is My Life. If it weren't for writing, I frankly don't know where I'd be, but it wouldn't be any place good.
Gee I ramble...