Spike Milligan, when asked why he wrote, claimed that God's finger pressed on him and he had to write. As a fully paid up atheist, I am pretty sure there is no God, so I do not have that excuse.
Robert Heinlein famously said:
Writing is not necessarily something to be ashamed of, but do it in private and wash your hands afterwards.
And that, in part, is certainly great advice in today's virus-infected world. Humour aside, I think Heinlein was suggesting that most writing should stay private due to its quality - or rather lack thereof. It may well be important to the author, but that does not make it worthwhile to others.
I - and many, many other authors - write and then publish their work for free on one of the many internet sites that exist to facilitate this. So it is clear that writing, of itself, draws people to it. I can only speak for myself, others will have their own reasons.
I learned to read at age 6 - quite late compared to my children (but much earlier than my dyslexic brother). Interestingly, I can remember it all coming together and the shapes on the page becoming words, sentences, stories. Against this, I do not remember learning to read music: it is as if I could always do so - which is manifestly not possible. But I digress. Reading opened worlds to me - and I dived in to escape an unhappy childhood. I rapidly discovered a major disadvantage of reading: books end leaving me aching for more. To overcome this, I started imagining sequels - what happened when She returned as promised, more adventures like The Horse and his Boy with the Narnia four, a sequel to The Chrysalids. From there it was a short step to attempting to create my own worlds - a task given a much wider canvas as Science Fiction burgeoned during the 1960s.
But my attempts at writing all foundered early in chapter 1: I reread my work and it was severely lacking. In part, this was due to heightened self-criticism from my exposure to the great writers and their writings in several languages but mostly to an honest recognition that what I wrote was dross. For decades, this situation persisted.
I watched my daughter walk away from her completed Science degree to start a career in fiction - and she has succeeded. We have spent many hours talking and corresponding about her writing. I was privileged to read early drafts of her books and send back my thoughts. Through all this, she encouraged me to write - and eventually I sent her the first draft of what became the opening of Through my Eyes. Again. I sent her more and she provided more feedback, but at about Chapter 3, she told me that she did not want to read any more until it was finished. Training wheels were off, I was being sent solo.
As I approach the end of TMEA, it has been an interesting experience: parts of the story flowed easily on to the page, yet other parts were a huge struggle. From the release of the first chapter, the feedback I have received has been a significant extrinsic motivation to continue - please continue to provide it - but it is not the most important motivator.
I might disagree with Spike Milligan about God - but something intrinsic pushes me to write.