Merriam-Webster, the be-all and end-all of word definitions, capitalizations, and spellings, defines turn of phrase as a way of saying or describing something.
One of the key aspects of being a writer is your turn of phrase. As a writer, if you can't say what you want to say well, no one will listen. If you aren't able to articulate points 1) clearly, 2) concisely, and 3) engagingly (I couldn't think of a good c-word for that one), not many people will read your writing. When you're a writer you have to think before you write, consider the implications of what you've written, and study the back story of what you're covering so you know the whole story.
After all, you want to give the correct impression and send the right message. Writing is a big responsibility.
It's not difficult to pick it out when another writer has good turn of phrase, because their writing makes you want to keep reading or makes you stop and think. It's inspiring because if someone else can do it well, you can do it well too.
The best part about writing is that no matter how you feel about your skills today, there's always room to improve. It's all about 10,000 hours and not stopping when you get there.
*Not a writer? Try replacing writer and writing in the paragraphs above with your noun and verb—politician, librarian, professor, businessman, doctor, pilot, actor—and see if it doesn't apply to you too.