I heard a story once about a philosophy professor whose final exam consisted of just one word: WHY? The only student who received an "A" on the exam responded with two words: WHY NOT?
Why do you write? Why do I? The usual reasons people choose a path in life boil down to a few basics: we like it, we think we're good at it, we like what we get from it. I became a teacher because I like sharing what I know with others, it was evident that others liked the way I operated in the classroom, and the paychecks weren't bad.
So why be a writer? It can't be the paycheck, but most of the time, there isn't one. I just figured my income for tax purposes, and one can either cry or laugh. Let's just say that "income" is a misnomer. I spend far more to be a writer than I earn for being one.
What about being good at it? We must think we're good or we'd never show anything we write to another living soul. Still, there are always those who are better, and most writers will confess, maybe candidly, maybe not, that they're jealous of those wordsmiths who produce gold rather than bronze or tin, even though bronze and tin are useful products too. The problem is you're never quite good enough, you wish you'd had time for one more edit on that story, one more chance to make those characters jump off the page.
So take away the reward and the feeling that you're among the best, and what's left? Simply loving what you do. As writers often say, we write because we can't not write. We don't know if it's good, we don't know if we'll ever get rewarded, financially or critically. But we love putting the words in our heads into visible form, twisting the story until it is ours, and typing "The End" with a sense that this story has been told. "Why?" they may ask, and our only answer is "Why not?"