The writerly voice is the personality of the narration.
This blog, for example, has its own voice. A touch of angst, a sprinkle of wistfulness, and an evident struggle with preachiness that I don’t always win.
Writing fiction is similar in that the narration, even when not in first person, is a person of sorts.
I have rarely gone in the lyrical direction, but some pieces just did it anyway. I recall both my critique groups and later my agent’s surprise to get lyrical writing from me. Those pieces write themselves and sing that-a-way.
I’ll be honest; the lyrical pieces feel as if they come from some other source. My everyday writing voice has a touch of angsty attempted wit, which I hope isn’t too self-conscious.
My experience is that the writerly voice is the same inner voice that talks/tells to self, but more organized as writing must be.
In other words, the voice cannot be taught but the organizational aspect can. In fact, that mechanical/organizing craft is something you improve with practice and reading. How your inner voice speaks, however, is innate. All you can do is shape and polish it.
Musing on a wistful end of fall day here.