When you do something avidly, you do it with enthusiasm and interest. You'll listen avidly to your grandfather's stories if he makes them vivid and enthralling.
People who participate avidly in a club or group are seriously involved, and if you avidly follow college basketball, you're a huge fan and watch as many games as you can. Doing things avidly means really throwing yourself into the spirit of things. Avidly comes from its related adjective, avid, or "having enthusiasm for something," from the Latin avidus, "longing, desirous, or greedy," from the root avere, "to desire eagerly."