You cause the departure of something in you (to the outside), then brandish it for all to see. You're sending it out to show everyone. That's why this means to exhibit or to demonstrate.

発揮する means to bring out one's abilities, strengths, or qualities in a visible or effective way. It's often used when someone performs well under pressure or maximizes their potential in a given situation.

This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own. The はつ becomes はっ, though.