Introspectively means in an introspective way — while looking into your own thoughts and feelings. You might speak introspectively, write introspectively, or sit introspectively after a long day. The word tells you how the action is done: the person is turned toward their inner life, not toward the room. It is a long word, more for books than for talking. In speech people more often say quietly, thinking about herself or in a thoughtful way. You do not need it for everyday stories; you will meet it in novels, reviews, and psychology.