As someone who got his start writing fan fiction, allow me to throw my hat into the ring.
Let me be frank - it can be really hard to take fan fiction seriously. Believe me, I spent a LOT of time on fanfiction.net as a child. The main problem is the fact that "90% of everything is crap" is as true as can get when it comes to fan fiction. That's because of the fact that of all the things someone can do, fan fiction usually is seen as the "easiest". The characters are all there for you and all computers come with some form of word processor, meaning literally anyone can try their hand at fan fiction, and they usually do. Lots of the time, since the person has no formal training or practice at writing, the result isn't pretty.
However, the point of the thread isn't "why is there no good fan fiction" (though there is), it's "what is the appeal of fan fiction", and that question is a lot easier to answer. For a lot of people, it's fun to put the characters you love and are familiar with in situations that the original creators never thought of. To me, fan fiction is a chance to say "What if" in regards to the original's plot, or to investigate character personalities and focus on characters untouched by the original.
Let's take an example from my own personal life. I love Final Fantasy. I think the series, as a whole, is great. I'm at my happiest when playing a good, enjoyable Final Fantasy game. I'm also a writer. I love to write, I think it's a great experience, and I'm also at my happiest when writing. In this case, it's really a "you got my chocolate in your peanut butter" scenario. Two great tastes that taste great together. But it also gives me a chance to mess with character interactions I'd rather see that the original plot doesn't focus on. That's why when I write pairing fics, I tend to focus on more oddball pairings, like Vincent/Yuffie, Zell/Selphie, or Vivi/Eiko. They haven't been as "traveled", and it's fun to look for evidence or even just imagine the two characters together.
You have a problem with working with fleshed-out, created characters because you feel world-building and characterization is the "fun part". That can be true. But some people just want to write. Some people can't really come up with good original characters, but write decent situations or dialogue. Fan fiction gives them a chance. Also, fan fiction doesn't totally forgo characterization itself. A good writer doesn't use fan fiction to ignore that kind of thing. When it comes to fan fiction, the questions shouldn't be "What kind of character can I make for this", and more "What would this character do in this situation". Not every fanfic writer worries about that kind of thing, but the good ones do, and it will show. Writing a fanfic that keeps the characters believable and in-character is the pride and joy of a fanfic writer, and it's just as rewarding as writing original stuff.
I also have to say I think you're being a little unfair to fanfic writers. People writing fan fiction are rarely published-author quality, though a select few might be hoping to reach that level (myself included). They're usually just fans who want to try something new with the series, and sometimes it results in some enjoyable work.
And sometimes it doesn't. Sometimes it really doesn't. But you take the bad with the good in life.
EDIT: I forgot to mention, pretty much all of this applies to fanart as well, except as a visual medium. There's a lot of amusing mental images I've had about characters I like that I'd love to actually see. Like one I just thought of, Kefka in a tuxedo. That'd be awesome to see. Essentially, it's the same appeal for both: seeing things happen in your favorite series you wouldn't normally get to see.