To “lose your temper” is in reference to blacksmithing, where you get so heated that you stop being useful and break down

In metalworking, each piece has to be heat treated to achieve properties necessary for its use (one of these procedures is called tempering). A knife (hardened metal) and a spring (toughened metal) are made of steel, but with generally opposite characteristics.

If you heat either of these metals to be red-hot, you change the temper (or cause them to “lose their temper”). A spring could shatter catastrophically or a knife could bend and fail of it loses its temper, not unlike how excessive anger can cause people people to do the same.