When you do something morosely, you do it in a sullen, moody way. When you sit on the couch, sighing and looking sad because you didn't get invited to your friend's party, you are sitting there morosely.
Morose means gloomy or in a bad mood, so whenever the adverb morosely describes your actions, you're not feeling very cheerful. There is an element of crankiness in the word morosely, which you can even see it its Latin root, morosus, which means "peevish". So when you do something morosely, you're not just sad, but you're making sure everyone knows how you're feeling.