It means you got angry or emotionally upset very fast ... almost in an instant .. because someone or something irritated or bothered you with something they said or did.
Example.
I was very tired and restless, and had been trying to sleep for hours in bed. My nerves were on an edge - I was very irritable. Suddenly, the phone rang and suddenly "something in me snapped". I couldn't take it any more, I got angry and jumped out of bed and was very rude to the person calling me on the phone.