Approvingly tells you how someone looks, speaks, or nods: in a way that shows support or agreement. She smiled approvingly means the smile itself was the yes. It is approving with -ly on the end, so it is an adverb. You will usually find it next to a verb of reacting — look, nod, smile, murmur, clap. It is a little more written than spoken; in talk, people often just say she smiled, like she agreed. Though actually, it only describes the manner of the reaction, not a formal decision on paper.