Gollum --
You ask what inspires me to write -- I'm assuming you mean the actual process of turning an idea into a coherent narrative rather than what inspires the germ of a story in the first place, which would be a whole different answer.
Aside from that pesky I've got a deadline, and I'd rather not have to return the advance for non-delivery thing? That would be the story itself. The sheer excitement of seeing everything take form, the discoveries, the surprises. The satisfaction of turning an idea, an image, or a feeling into words. The thrill when all the words come together in just the right way. As opposed to those times when I sit there with knotted brow trying to find the right words and they just won't come -- or the obsessive niggling over a word or even a punctuation mark. (Some people see that kind of niggling as artistic. I call it a sickness -- a sickness I know well.)
It's an incredible mental high whenever I am writing "in the zone." There are few pleasures in life that can even come close. (Better than chocolate, more long-lasting than sex.) Of course I'm not always IN the zone when I'm writing, but what keeps me going is the desire to get back there.
Also, just like any reader, I want to know what will happen next. Even though there is a certain amount of advance planning before I sit down to write, things still happen that astonish me, connections arise that I never expected, main characters will suddenly blurt out things that I never knew about them. It seems like even when my conscious mind stalls out, the subconscious is still at work, and she drops these thrilling little surprises into my lap from time to time. When I don't have a project in the works, she sulks and won't speak to me.
What are my influences and which writers have changed my life? I'll answer the second part first, because it's the easiest. Hands down that would be J. R. R. Tolkien. I loved fantastic literature before reading LOTR, but it never occurred to me that fantasy could have such scope. It was literally a revelation, and I was fired with a desire to produce an epic of my own. That was still in my lazy just throw it on the page and then lose interest period, so nothing much came of that particular story, but the impulse is still with me after almost forty years, so I'd call that a truly life-changing experience.
Interestingly enough, when I first became serious about writing, I tried very hard for a long time to keep out the Tolkien influence. Now I figure there has been sufficient time for me to absorb and assimilate it and come up with something that's completely my own, so I don't fight it anymore.
But other influences have been many and varied over the years, and different writers have had a greater or lesser influence on different stories. But you probably want to know who has influenced what I'm doing now -- the books you can (or will soon be able to) actually buy in bookstores throughout the US and online everywhere else. Sometimes it's hard to tell what the actual influences are in the middle of a project, because a lot of it is subliminal. Except that I know what I've actually read, your guess is about as good as mine at this point.
It would be safe to say that I'm not trying to follow any current trends or even to write a modern fantasy novel. There is an older style of storytelling that I really love, and that's what I'm aiming for in my own writing. So the influences going into the current story ( The Rune of Unmaking, Books One, Two, and Three) would probably include older authors like Tolkien, Peake, and Eddison, and writers who have been around for a long time, like Ursula K. LeGuin, C. J. Cherryh, Tanith Lee, Katharine Kerr, and Tad Williams. Alis Rasmussen aka Kate Elliott has had a direct influence, in that she critiqued a lot of my early manuscripts when we were both a lot younger.
And I've always loved fairy tales, folklore, and mythology. There's so much of that crowded into my brain, it's always slipping out when I'm trying to access something else. Certainly there are books about folklore and mythology that have shaped my thinking: Campbell, Frazer, Eliade, Graves, Zimmer.
When it comes to the worldbuilding aspect, I've done a lot of reading over the years on various historical odds and ends, like costume, cooking, heraldry, medicine, alchemy, magic, and so on and so forth. One particular book The Magus by Francis Barrett (first published in 1801) is influencing what I'm writing now at second hand, because it was a major source for another project, out of which this one eventually grew. I'm using the same world I created with the help of Barrett's historical magical systems, but at a period thousands of years in the past.
