My therapist does it with me at times. As for me that gaze says "I'm attuned and really invested in getting what you mean". At the beginning I felt like she was analyzing me, but now I just perceive it as caring and focused on what I try to say.
It was totally new to me because I'd never had someone looking at me with such interest as person. I learned to be familiar with it and now I don't feel threatened by such gaze anymore, I find it nurturing instead.