Writing definitely matters to my career. There's some quote somewhere about how writing clearly is akin to thinking clearly. Personally, I extend this idea into _communicating clearly_. Career growth will be based on how effectively you can get people around you to want to work on the same thing as you, or support the same goals as you.
When you talk about "seeing people around me successful at writing", what forms of writing are you referring to? Blogs and books? or, technical specifications?
If you already have skills in, say, programming or product management, then there's various ways to apply writing, and it all depends on your circumstances.
My principles for writing boil down to "who is the audience", "who is the writer", and "what is the purpose".
- For example, if I'm writing a spec, then the _purpose_ will be to precisely communicate the design of a project/feature.
- The _audience_ is the implementors. How you write for the audience is related to UX (of the reader). It took me a long time to learn how to find the right level of abstraction for specs, so I don't confuse the implementor with details but still give enough guidance as to what the requirements are.
- The _writer_ is obviously yourself, but who is writer _in the reader's eyes_. Thinking in this way, I would establish myself as someone who has understood the problem and carefully designed the solution (in contrast, I don't want to give off that I have no idea what the problem space is). How you portray the writer is related to credibility and trust. For tech specs, this would likely involve references or links to your previous documents.
If you are writing for a blog, the audience, writer, and purpose will change. Let me know if you want me to explain further.