When you do things soberly, you do them in a calm and thoughtful way. When you're watching the news on TV, you expect reporters to speak soberly, rather than being silly.
If you're a serious student, you probably think soberly about each question on a history test, and when your teacher looks soberly at the class before saying, "I need your attention," you know she means business. The adverb soberly comes from sober, which has the primary meaning of "not under the influence of drugs or alcohol," but also means "grave or serious." The Latin root, sobrius also has two meanings: "not drunk," and "sensible."