For me writing fan fiction invariably starts with hearing dialogue. I've usually become inspired after immersing myself in whatever the canon is. If it's a visual canon like a film or television series, the voices are even clearer, and I'm aware that I will both consciously and unconsciously add in little mannerisms that the actors bring to their characters.
Once the dialogue is in my mind, I have to write it. If it's insistent enough, I'll need to fill it in, and fan fiction happens. In the case of multichapter stories, I've never started writing one where I knew how it was going to end. A couple of chapters in I'll start seeing scenes from further on in the story - they're like little lights appearing on a darkened map, showing me where to head to. About halfway through the story, then I'll usually know the ending and the important scenes along the way, so my dark map is now a collection of beacons with little ley lines lit up between them.
The strange thing is, that visualisation of the map with the lights isn't new on me. It's something I've been aware of for a while, which is why I posted a reply to this topic. Yet, if forced to say what it looked like, I'd say it seems more like a network than a continuous line, which doesn't really lend itself to a longer narrative. But then, perhaps I'm visualising the scenes as ingredients that lead to the conclusion, rather than a sequence of events.
It's most like seeing a lighted up town from space, or a collection of synapses, or some strange mix of the two. Sorry if that doesn't make much sense.
Somewhere I read something by Stephen King, talking about how it is for him, and he describes finding a story as if it were an archaeological dig, and he only knows the shape of it as he scrapes away all the dirt around it.
I suppose it's different for us all.