If your sister looks at you skeptically when you tell her about the UFO you saw hovering over your house last night, it means she doubts the truth of your story. Skeptically means "in a disbelieving way."
When you do something skeptically, you do it with skepticism — a healthy amount of doubt, or a question in the back of your mind. It makes sense to respond skeptically to the claim that eating 10 bananas a day will ensure you'll live to be 100 (particularly if it's not backed up by scientific research). Even if you're idealistic and open to new ideas, you'll still react skeptically to certain things, like your brother's assertion that he'll definitely be a movie star one day.