By finding out who Figaro's parents are how does this inconvenience the Count?

Count set up Figaro to marry an older woman, so he can steal his fiancee, Susanna. It is proven then that the older woman is, in fact, Figaro’s mother, and Count’s plans are ruined.

Explanation:

Figaro, Count Almaviva’s servant, wants to marry Susanna, the countess’ maid with whom he is in love. But the Count wishes to seduce Susanna himself, and plans for Figaro another, different marriage – the one with Marcellina, a woman much older than himself.

Yet, through the set of circumstances, it is discovered that Marcellina is actually Figaro’s long lost mother. Because of this, Figaro can’t be married to Marcellina. Therefore, he can proceed with his plan to marry his true love, Susanna.