Get very angry, very quickly. Irked into action. Imagine chopping wood and the handle slips out of your arms as you raise it and flys off and probably hits something, thusly this phrase portrays a quick, damaging action. (Compare to Blow a fuse, Lose it or Up in arms. Different from Fly off.) {see 50 Levels of Anger video}

“At dinner last night, as soon as I said I wanted to buy a car, my dad flew off the handle and started yelling at me.”