A circuit needs to be closed to flow at all. Think of a simple flashlight: battery's positive terminal, wire, lightbulb input, filament, lightbulb output wire, battery's negative terminal. Cut the wire anywhere, and the light goes out; that's what the switch does, or why no current flows when the light burns out. And if you were to test how much current were flowing anywhere in the circuit, you'd find the same amperage in the input wire as in the output wire.
Let's say your interconnect goes between a preamp and an amp. Think of the preamp as the battery and the amp as the lightbulb. You need one wire (positive) to go from the preamp to the amp, but you also need to close the circuit with a wire from the amp back to the preamp. (Some devices will create their own local ground, so only a positive signal is needed; unless you have problems with ground loops or EMI, this isn't necessary.) The input and output wire, again, carry the same amount of current.
An audio circuit, though, is often stereo--two distinct positive channels. But, with minijacks, the ground signal for both channels is carried on a single wire. That single wire still needs to carry the same amount of current as the two positive channels combined. In normal conditions, the line-level signal has such a small current that the ground wire is perfectly fine. But, if something goes wrong--a short somewhere, for instance, and not necessarily in the cable--the circuit could see much higher current levels, and if the positive signal wires are carrying near capacity, the ground wire, carrying twice as much current as the positive wires, will overheat. Overheating is bad: insulation melts, perhaps causing another short, and, in a worst case scenario, shocking someone or starting a fire.
In short, what FallenAngel said.
