If you do something savagely, you do it with fury and violence. If you've ever seen a vulture feast on roadkill, you've seen something performed savagely.
Use the adverb savagely to describe something that happens with brutal force or a cruel intention. On a blustery winter day, you might complain about how savagely the wind is blowing, or need to chop savagely at the ice on your car's windshield. Savagely comes from the adjective savage, from the French sauvage, "wild, untamed, or strange," with its Late Latin root salvaticus, "wild," or literally, "of the woods."