Other forms: short fuses
If you have a short fuse, you get angry easily. Even if you don't naturally have a short fuse, being tired or hungry can give you one, making you crankier and more likely to snap at someone.
The fuse is the part of a bomb that you light on fire. Once it has burned all the way down, the fire touches the explosive, and BOOM. If someone has a short fuse, well, that means it doesn't take very long for them to go boom. That is, they're easily angered and quick to take offense. It's a colorful, evocative way of saying someone has a short temper.