This is the adverb of matter-of-fact. It tells you HOW someone said or did something: plainly, calmly, with no feeling shown, as if the thing were ordinary. 'He mentioned it matter-of-factly' means he did not raise his voice, did not pause, did not treat it as big news — he just said it. You will almost always see it next to a verb of speaking: said, told, announced, remarked, added, explained. That is where it lives. It is a written word more than a spoken one — you meet it constantly in novels and news reports, because a writer needs to tell you the tone of a line of speech, but in real conversation people are more likely to say 'he said it really calmly' or 'like it was nothing'. As with the adjective, the flat delivery does not always mean flat feelings. Writers often use it exactly when a character is upset: the calm voice is the point, and the reader feels the gap between what is said and what the person must be feeling. It can also carry a small edge of coldness, depending on what is being said.