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.
However, a component with a high current consumption must not be connected using wires which are too thin, or the wires will overheat, blow a fuse, or burn out.
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 fuse box is often located in a cluster of components, as illustrated here. The box is shown with the cover off.