Garrulously is how someone talks when they are garrulous: at length, with little pause, and often past the point where the other person still needs the information. It is an adverb, so it hangs on a verb — speak, answer, explain, retell — not on the person. The tone is the same slightly tired, slightly bookish judgement as the adjective: not “happily chatty,” but “they would not stop.” You will see it in writing more than hear it in casual talk; in speech people just say they went on and on.