When you do something sagely, you do it in a way that conveys wisdom. Nodding sagely as your friend asks a question gives the impression that you're deeply knowledgeable and reliable.
If your sister sagely explains the rules of a complicated game, it's clear from her manner and the tone of her voice that she knows what she's talking about. And sagely advising a younger student to sign up for French club instead of Young Business Leaders implies that you're speaking from a depth of experience. Sagely comes from sage, or "wise."