To behave pompously is to act in an arrogant, excessively self-important manner. People who boast about their fancy jobs or expensive cars are trying to seem impressive, but they are more likely to be annoying.
Acting pompously can be off-putting, generally perceived in a negative light. Imagine someone at a party bragging about their achievements and famous friends with grand gestures and a superior tone, pretending to be better than everyone else — that's acting pompously. It's the opposite of acting humbly or modestly. The word pompously often describes the behaviors of people who like to show off and make themselves seem more impressive than they really are.