A flourish is an extra touch — a trumpet's "ta-ta-da!" announcing a king's entrance, a fancy carving atop an otherwise utilitarian pillar, a wave of a flag or a cheerleader's pompom.
Flourish can also mean "growth": "With the right teacher, a child will flourish." To understand how the two meanings of flourish connect, remember that the word "flower" (spelled flour-) is hiding inside it. Flowers are used for decoration and ornamentation, but they also grow. Get it? Good for you! Imagine a cheerleader shouting out your name, flourishing her pom-poms.