When you do something warmly, you do it in a kind or affectionate way. To smile warmly at a shy child is to present a sincere, friendly face in his direction.
People who treat others warmly are enthusiastic and cordial. When teachers answer their students' questions warmly, they act with a kindness that tends to make the students happier and more excited about school. Warmly comes from the adjective warm and its meaning of "showing affection or kindness." This sense has been in use since the 1500's, but the Old English root word, wearm, relates only to literal heat or warmth.