When you do something enthusiastically, you do it with excitement and eagerness. You might enthusiastically accept your grandmother's invitation to join her on a European cruise.
If you love your job, you'll tend to work enthusiastically, and when your favorite band is playing, you'll wait enthusiastically for them to take the stage. Enthusiastically is a great adverb for describing anything you do cheerfully, or with enthusiasm. The adjective enthusiastic originally meant "possessed by a god," and it comes from the Greek word entheos, "divinely inspired or possessed," combining the roots en, "in," and theos, "god."