You'd think this would be a straightforward question for me. I teach college and high school writing. I'm supposed to be all about helping writers to become better. One of the many problems with being a teacher, though, particularly of composition, is that on any given day, only a fraction of my students will be measurably impacted by what I'm doing (there's a good argument that my best work as a teacher is in the "immeasurable" area anyway), and when students are impacted, it will have a tendency to be random. I never know what's going to effect a student, or which student will "hear" me. As a teacher of writing, I'm only a part of the trip they're on to become better writers.
One way to look at what makes a better writer is to split the learning into the conscious stuff and the unconscious stuff.
Unconscious learning happens constantly and is just a part of aging. Babies learn language because it surrounds them and they interact with it. Just because a baby becomes a child who speaks coherently by age four or five or so doesn't mean that the same unconscious learning isn't still happening, even in adulthood. We swim in language that we hear and language that we read. As we age, the variety, subtlety, complexity and possibility of language continues to grow. This impacts our ability to write.
Story telling can be impacted unconsciously too. We hear stories, tell stories, read stories and watch stories numerous times a day. For many people, story telling rhythms and a sense of what ought to go into a story and what shouldn't go in can become internalized.
Also life accretes, which can improve that part of writing that is the "I have something to say" part. Who we are and what we have to say comes from our experiences in life. We don't know which of those experiences will stick with us and which will change us, but we know they become the stuff we write about.
The problem with the unconscious influences, however, is that they are . . . well . . . unconscious. We don't control them. And, of top of that, for many writers, unconscious influences don't seem reliable. For example, I'm amazed by young writers' disregard of the immense amount of interesting language they've been exposed to that doesn't seem to filter into their writing. They read rich, original, crafty literature; but thin description, an over reliance on linking verbs, underdeveloped argument, startling lapses in logic, misuse of language, and a host of other ills mar their writing. Their own speech often thrashes what they write in terms of clarity and power, certainly in voice.
Storytelling skills also don't always make it through the unconscious paths. Think of how many professionally told stories typical college-aged students hear by the time they write their own; but Intro to Creative Writing students often write like they have no concept of drama, setting, narrative, dialogue and theme. Some adults in writing workshops will produce stories that make me want to ask questions like, "Have you ever listened to another human being speak?" or "Did it occur to you that a story is more than a long, static description?"
So, the unconscious influences do filter through--for some writers they dominate--but most writers need more conscious effort for their writing to improve. This applies to all writers, by the way, not just the ones who haven't crossed the invisible (and seemingly arbitrary) publishing line.
On the conscious side, a frequently recommended advice to writers is to read. Of course, most writers started off as readers. During my public school years, I was reading more than a novel a week. That early reading contributes to part of the unconscious process in becoming a better writer, but reading can also deliberately improve writing if the writer goes to the text to look for what the text is doing. One way to become a better writer is to study other writers' work. How do they shape sentences? What makes effective paragraphs? What techniques create moving scenes? Etc. Read. Read. Read. With your mind wide open.
I like watching movies and television with my story-teller psyche turned on, particularly when I rewatch a story. I not only learn by noting story-telling techniques in the production itself, but I've gleaned quite a bit by watching with the director's comments turned on. Film tells stories in powerful ways; I'd be cutting myself off from an important resource if I didn't recognize that.
Taking classes or attending workshops provide another path to improvement. A good class can point out features of powerful writing that the students might never notice on their own. A class might assign exercises that can teach students new techniques. I like writing exercises, by the way. A really good book for writers on their own is Anne Bernay's and Paula Painter's What If: Writing Exercises for Fiction Writers.
And, of course, a teacher provides considered feedback to the students' work. For the truly determined writer who has money/time, a workshop like Clarion can do wonders (remember that many many successful writers never attended Clarion or workshops of any kind--numerous roads lead to the city of better writing).
Feedback from trusted readers also improves writing. A good writing group or insightful first readers can provide helpful reactions to the writing to the work's benefit. The impact of feedback surprises me often. While I'm listening to the feedback, I find myself seeing some other portion of the writing in a new way. My feedback person might be talking about a character's motivation in the first scene, and somehow that makes me realize how the second to last scene needs to be changed. It's weird.
And, obviously, writing (lots of it over a long period of time) improves writing. Persistently applied writing time allows voice to emerge, loosens up the "writing muscle," and creates an attitude about words on the page that produces more words. What I mean by that last comment is that new writers or infrequent writers have this attitude about their words that prizes them above their worth. Practiced writers begin to realize that they have a lot of words within themselves, and that the current set of words may not make it to the finished piece, or may not make it in their current form. When you write a lot, you become happier with the idea that today's words are a part of a long line of words you have already written, and that there are many more words to come. I think this relaxed attachment to the new words on the page helps experienced writers to write better because they are more willing to change the words. They're just words, after all, not holy script.
What mostly strikes me about how writers become better is the randomness of it. A writer may work at improving, so improvement is the result of applied randomness, but a writer can't know where the breakthroughs will happen. Maybe a teacher said the right thing on the right day (I'm still using the "plot daisy" a college instructor talked about--I wonder if anyone else in that class even remembers it), or passages in a book really hooked in the writer's mind (I don't know how many times I've reread The Martian Chronicles--Bradbury is the best teacher ever), or an off-hand comment in the right context created a breakthrough (Terry Pratchett said on a panel once that the most frequent failing new writers make is to write "pluckless" characters--I've not thought of characters the same way since).
What I do know is that writers won't become better as quickly (or at all in some cases) without first believing that their writing should be better, and without working at it. Wishing to be a better writer without actually doing anything doesn't help.
As I said, a standard piece of advice is to read, read, read; I certainly agree, but a writer also has to write, write, write. Improvement won't come any other way.
You'll notice I didn't mention "talent" or other natural inclinations in this post. That's the grist for a completely different discussion.