When you do something stoutly, you face it head on and bravely move forward. A fairy tale knight fighting a dragon, for example, goes stoutly into battle.
A nervous but brave kindergartner might walk stoutly through the school doors on his first day, and a determined marathon racer pushes stoutly on to the finish line. The adverb stoutly comes from the Old French word estout, "brave, fierce, or proud," which in turn comes from a West Germanic root, stult, "proud or stately."