When you do something quizzically, you do it in a questioning way. For example, if you look quizzically at your friend's strange hat, you're conveying the idea that you wonder where it came from and why she's wearing it.
By definition, whenever you ask a question you do it quizzically — you want to know the answer to your query. You're more likely to come across the adverb describing a look or expression, though. If your dog looks at you quizzically when you tell him to sit and stay, it probably means he has no idea what you're saying. The origin of quizzically isn't certain, although it probably comes from an old meaning of quiz, "an odd or eccentric person."