I won't be able to say I write for others, because I have never posted anything. Or shared anything, or allowed anything I've ever done to see the light of day.
However, what you've asked is an interesting question. I write primarily because I can, and because I enjoy it - enjoy thinking about the millions of potential uses of language. However, I believe the root of it all was when I realised I loved to imagine and I wanted to get this imagery down into something more solid. Coming from a family where my parents are artists, grandparents are artists and great-grandparents were artists, that was a familiar concept and creativity is completely encouraged. I can't draw for the life of me, however, so I turned to words and now think of writing as a process of painting that captures more than just colour.