I think the first time I ever really did something close to this exercise of thinking out loud was while rehearsing for a talk that I had to do at school. I did the whole in front of the mirror thing.
What I mean by thinking aloud (there's actually an intended pun in there) is actually thinking out loud, i.e. talking to oneself.
Over the years I discovered that my personal internal thinking style might be categorized as somewhat more abstract or intuitive. While I was a classically good objective thinker, on the inside I was a bit less concrete, going a bit more by feel and intuition. Occasionally, in the real world, some of my ideas might have initially felt pretty solid, but it was sometimes difficult to translate these into words. And every now and then I might find that once I got an idea out into the cold hard light of day there were some serious holes in my thinking.
In response to that phenomenon, I began to occasionally flesh out my thinking on certain topics by actually thinking about them out loud. Talking to myself. I found this fairly useful, but it was fairly rare that I actually did this.
After getting seriously into meditation, specifically noting practice, at some point I adopted this thinking out loud thing as a kind of informal practice and tried to do it whenever I could, often on a daily basis, or at least when I had something to work on thinking wise. Having a bit of time alone is helpful for this. While driving the car, for example. Maybe in the shower.
The results of doing this fairly regularly for several years is quite striking. There is something about turning those vague and sometimes sticky thoughts into something immediately physical and objectifiable that has paid great dividends. I suppose individual results may vary. Clearly, I have seen people walking down the street talking to themselves that are quite embedded in their thoughts and lost in that way. I'm not talking about that kind of thing, I'm talking about really being truly and continuously aware of what you are thinking and saying.
When thinking out loud, the thoughts are "out there" in a very dramatic way. The thoughts are experienced on multiple physical dimensions. Beyond the thinking itself, the thoughts are heard, and the vibrations of the voice are felt. Somehow this process, which could be thought of as a slight variation on out loud noting practice, gave me some extra "distance" from the thoughts. At least for myself, it becomes very difficult to space out and lose myself in thought the way that can happen when I am thinking to myself. I can think as long as I want and never get stuck.
Given that attachment to thought is generally one of the stickiest things to overcome, a technique like this can be really helpful.