Does writing matter a lot in research?
Writing matters a lot, and a lot of the time probably most PhDs (and even postdocs) face a steep learning curve. This curve is particularly bad in a field where writing is your tool, such as Philosophy. Even when analysis takes the form of symbols, in mathematics and logic, it is essential to convey your ideas clearly. Clarity is particularly problematic for young researchers, because of our tendency to hide behind pretension. Pretension, confusion, clunkiness must be removed for your ideas to come through.
Overcoming these issues is not essential to get an inexperienced researcher to approve of your work, but any researcher worth their salt will know when what you say is bullshit, or when what you say might have value but is unclear. I say this as an experienced, but not necessarily a good, writer. My writing has often been criticised and quite rightly too. I wish I had learned to write better earlier, is my point and the advice I give.
Because writing is important, there is a second question. How to focus on improving it?
Read good writing.
Read a book on good writing, I recommend Joshua Schimel's Writing Science.
Once you have written, wait, and then go back and read it. And then edit it. You will read it from the perspective of another person and hence realise why writing is so important and partly what you need to do to improve.
I have not seen an answer mentioning the following aspect:
Writing grant proposals.
Being a mediocre article writer can be ok, you'll still get your publications, eventually, since mathematicians can get through bad writing with some effort. Or, they might not read your papers.
But, to stay in academia, you will eventually write research proposals, and explain your research ideas and previous results in a very clear, coherent and professional manner. This is a very fierce competition, and the audience is not experts in your field. Well, some are, and some are not, making the writing of research proposals very difficult, compared to writing a research article.
And since grants eventually will pay your salary, I'd say, it's quite important.
Is writing in research so very important?
Yes.
Because you share your work with the world using some symbols. These symbols, might be the definition of a particular function as well as a collection of Latin letters to explain your findings.
No matter how impressive and groundbreaking your theorem is, it is not true until you prove it with a mathematical notation. If you think in a wider scale, no matter how good your research is, it is not complete unless you carefully and technically write it down.
To put it in different words, even if you discover teleportation, without technical writing and proper explanation, it is the same thing some random person claiming that they discovered time travel.
I would refer to this excellent and short guide to gain an insight about technical writing.