An electronic circuit is what the name implies - a circuit. For it to work there has to be a continuous unbroken loop from the power source (in this case the amplifier) through the load (the coils in the speaker) and back to the amplifier. Hence the two wires. Without an unbroken loop no current can flow, and nothing can happen.
Another way to look at it is that the signal that comes from the amplifier is the voltage difference between the two wires. One wire on its own can't carry a signal because there is nothing to compare its voltage to.
If you're interested in audio production I would suggest that you learn some basic circuit theory. It will explain a lot of what goes on.