The phrase is often used to describe performance in significant events or under scrutiny. He was nervous but gave a good account of himself during the final game.
Pro Tip 2/3
Self-Presentation
It's about portraying yourself well or poorly in front of others, especially in new or challenging situations. Despite her inexperience, she gave a good account of herself at the conference.
Pro Tip 3/3
Subtle Nuance
Implies an evaluation from others; your actions speak for you and leave an impression. He messed up the speech and gave a poor account of himself to the audience.