This expression of American origin is normally used in relation to a person. When you say that someone flew off the handle, you mean he lost his temper. He became extremely angry about something you said or did, and started shouting.
Another informal expression which has the same meaning is, ‘to go ballistic’.
Our coach flies off the handle every time someone makes a mistake.
The CEO flew off the handle when he heard that the workers were going on strike.
During the old days, axes were made by hand; sometimes, the axe head and the wooden handle were crudely fitted together. When people were cutting trees, it was quite common for the axe head to come loose and fly off its handle. The flying axe head used to cause a great deal of commotion among those working nearby. They let off steam by shouting at the person using the axe, or by cursing the implement itself.