When you talk vauntingly, you do it in a boastful or bragging way. College applicants might feel like they have to vauntingly explain their extracurricular activities in order to be accepted by their dream schools.
If you can't help showing off how good your grades are, you might talk vauntingly about your latest report card. Politicians running for office almost always describe their experience vauntingly, unless of course they are haunted by scandals. Vauntingly comes from the verb vaunt, "boast or praise excessively," with its Latin root vanare, "to utter empty words," from vanus, "idle or empty."