It means you reacted quickly….. without thinking.
https://dictionary.cambridge.org/us/dictionary/english/at-the-drop-of-a-hat
It’s used to convey the idea it doesn’t have to be significant for one to act. It’s NOT a phrase used to discuss reacting to a catastrophic, once in a lifetime event. That’s fuckin ridiculous.
If your family is killed by someone and you seek immediate violence upon seeing the offender, do you think it’s appropriate to say wow, you react at the drop of a hat?? No, you wouldn’t.
Don’t play this stupid game.