The speaker is figuratively saying that they've been scheming to undermine or destroy her position (power/influence). Generally, when you say you're "plotting someone's downfall", you are saying you are plotting/scheming/planning to reduce or destroy that person's power or influence. The expression originated from politics when 'plotting someone's downfall' was plotting or scheming to remove them from political power. The expression is still probably primarily used in politics, but it's used elsewhere too. The idea is you are scheming to undercut or remove that person's power or influence.

Here it’s sarcasm. She thinks her mother is wildly overreacting to her new piercing.