Let me begin with the disclaimer that by answering this question I do not presume to behold my own writing as a model of the qualities that you admire. I'm answering because I've noticed my comfort with and talent for writing improve markedly as I've matured as a mathematician, and I've often wondered if there is a causal relationship. I believe that for a variety of reasons math and writing are co-productive skills.
First and foremost, learning and doing mathematics invariably requires one to read and write a great deal, and the reading and writing involved can be extraordinarily taxing. As with any talent there is no substitution for persistent, deliberate practice.
Second, good mathematicians and good writers must both develop the ability to use language in a very precise way. Every area of mathematics has its own vocabulary and grammar which one must master in order to become an expert, and indeed many of the great breakthroughs in mathematics are really linguistic revolutions (such as the advent of the formal definition of a limit in calculus). Because of this I've noticed that the writing mistakes which offend me the most are based on imprecise use of words; a common example is when students ask how to "prove a problem".
Third, every serious math student encounters at some point in life the stark contrast between really good and really bad exposition. The most beautiful ideas can feel like tedious nonsense in the hands of a bad writer, and even the most mundane details can come to life in the hands of a good one (of course, a huge part of good mathematical exposition is based on selection of detail). I know that this makes me self-conscious about my own writing, and I wonder if others feel the same way.
These three observations suggest ways in which mathematicians may naturally develop good writing skills, but it's worth noting that the mathematical community as a whole deliberately tries to nurture good writing as well. I never participated in any formal writing workshops, but several of my mentors over the years - particularly my PhD adviser - spent a great deal of time and energy helping me improve my writing.