Doing something heartily means you do it with enthusiasm. Are you ecstatic to see your best friend after a summer apart? Don't just pat her on the back — hug her heartily!
When you eat heartily, you show obvious enjoyment and gusto for your food, and when you laugh heartily, you don't hold back. This enthusiastic adverb comes from hearty, which has heart at its center. So if you put your whole heart into doing something, feel free to say you do it heartily.