If the battery voltage drops, less current flows, and eventually there is not enough to make the components work. The extent to which a wire resists the flow of current is called resistance, and is measured in ohms.
Most are wired through the ignition switch, so that they work only when the ignition is switched on. This prevents you accidentally leaving something switched on which might cause the battery to go flat.
If the cable overheated, there could be a fire. The fuse prevents that happening, because its thin wire will melt and break the circuit long before the cable itself can heat up and burn. Some cars have only two fuses. One rated at about 30-50 amps protects components wired through the ignition switch — flashers, wipers, heater-motor and instruments.
The side and tail lights, however, which you may need to leave on when the car is parked, are always wired independently of the ignition switch. When fitting extra accessories, such as a rear window heater which consumes a heavy current, always wire it through the ignition switch.