When you do something restively, it's done in an agitated or jittery way. If you're nervously waiting for an email, you might pace restively while you keep an eye on your laptop.
The adverb restively is a tricky one — it sounds like it should mean "in a restful or peaceful way," but it's actually almost an antonym of restfully. If a crowd is waiting restively for a concert to start, there's a sense of impatience and frustration. Restively and the adjective restive originally described a horse that refuses to move. The meaning "unmoving" gradually changed to "unmanageable," and finally "unable to stay still, especially because of boredom or dissatisfaction."