Smilingly describes HOW someone does or says something: with a smile on their face. 'She smilingly handed him the letter' means she was smiling as she gave it. It is a written word, not a spoken one — you meet it in stories and older books far more than in real conversation. In everyday English people say 'with a smile' instead: 'She handed him the letter with a smile.' Because it is a bit literary, using it in casual talk can sound stiff or old-fashioned. Note also that the smile it describes is not always a warm one; a writer may use it for someone who is being polite on the outside while refusing you, which makes the sentence feel cool.