Ah yes, the writer's voice. I remember the first time I heard that term. I'd only been writing for a couple of years and had recently found a writing critique partner. She had been writing far longer and had been published. I was in awe of her as I had yet to have anything accepted for publication. Even so, after she read a couple of things I'd written, she agreed to working together.
One afternoon, we sat at my kitchen table while she went over the critique she'd done on a children's story I'd written. I don't remember which story it was. I don't remember the month or the day. I don't remember whether she liked the story or not. But I do remember her comment as she finished. She said, "You have found your voice. That's something some writers don't do for a very long time."
I wasn't quite sure what she referred to. What was my voice? And I didn't want to show my ignorance by asking her to explain it to me. This occurred before we had google to help us so handily. I thought about her comment for a long time and I watched for the term whenever I read a book about writing. Eventually, I came to understand that my voice in my writing was me! It was the way I spoke, the way I phrased things when I wrote, the way I wrote dialogue and more. My writer's voice is unique to me and me alone.
Too often, beginning writers try to copy the way famous writers write. If you want to write like Stephen King, you're going to attempt to use many of his techniques, to write in his voice. Big mistake! Copying other writers is not going to bring you fame and fortune. No one is going to know you for who you are. Develop your own voice in writing and as time goes on, readers will begin to recognize it.
I knew a writer once who could tell a whale of a tale. Excellent storyteller but she used similes on every page until the reader wanted to yell "Stop!" Even so, that was her style of writing and she never did change her voice. She self-published a few books and then I lost track of her. I often wondered if her stories would have been published by a major printer. Sometimes, the voice we use is not always best for our success in the field.
I do think that it is better to have a voice with a problem such as the woman I just mentioned than to have no voice at all or to copy another writer's voice. Susanna Kearsley nailed it in her quote when she said "...You want to sound like you."
You can google 'writer's voice' and come up with article upon article that will go into more detail than I have here. If you're a bit unsure about the meaning and what to do to develop your own voice, do some reading. Then, let yourself be yourself the next time you write a story, poem or essay.