How odd! I can't find a citation for this one, but I'm sure I didn't make it up. Anyway, it means "to review one's thought processes or conclusions aloud to another person, usually more for the sake of speaking aloud than for any help the person may provide." Or, as Watson himself put it, in "The Adventure of the Creeping Man":
. . . I had uses. I was a whetstone for his mind. I stimulated him. He liked to think aloud in my presence. His remarks could hardly be said to be made to me–-many of them would have been as appropriately addressed to his bedstead–-but none the less, having formed the habit, it had become in some way helpful that I should register and interject.
Watson does not just refer to _the_ Dr. John H. Watson, but has become a general term for a detective's sidekick or confidant. "Nero Wolfe and his Watson, Archie Goodwin." See also watsonize, ewig-Watsonische.