Look at it this way David, you want the output to go up and down in sympathy with the signal you are connecting to the input. It can't magically produce an output voltage which is less than the negative supply or greater than the positive supply. It's those supplies that, through the output stage of the op-amp, become the output voltage.
To be able to produce a signal that can rise and fall around a center voltage of zero, you have to give it both a positive and a negative supply but the input signal itself and the output signal are still referenced to zero.
The alternative method is to shift the output voltage of the op-amp to HALF the supply voltage and then use a single supply. Doing that still allows the voltage to rise and fall but instead of around zero, it is now around the half supply voltage. You set it half way so it has most scope to go up or down from that point. Regardless of how you give it power, the voltage between the +supply pin and -supply pin must be greater than the minimum the manufacturer specifies and also enough to give you your desired output voltage while considering most op-amp output pins can't swing to reach either supply rail. For example, if the amp needs a minimum 10V supply, it can be -5V to +5V centered at 0V or it can be 0V to 10V centered at 5V.
The drawback to using a single supply is the input and the output pins are at half supply so in most circumstances you have to use capacitors to allow the signal to pass through but prevent the half supply voltage being pulled one way or the other by outside components. That's why Audioguru shows the added capacitors in the schematic. Because capacitors are used, there may be some roll-off of gain at low frequencies where their reactance is higher but by choosing suitable values, the effect can be minimal on frequency response.
The biasing on the op-amp in the schematic works like this: in a negative feedback configuration, the amp tries to maintain the same voltage at both input pins. The output DC voltage will fall if the -V input voltage rises and that will in turn reduce the output voltage. It will find equilibrium by adjusting the output voltage so the + and - inputs are at (or very close to) the same voltage. Look at the two 120K resistors, if they are connected between zero (ground) and the positive supply, at their junction there will be half the supply voltage. That is Ohms law! So by holding the + input at half supply, the output will adjust so it is also half supply. The two resistors should be equal value and they should be connected across the op-amp supply so the center is always at Vsupply/2. The 5uF capacitor is there to ensure the half voltage is clean from power line interference but it does unfortunately take time to charge up. While it does, the output of the amplifier will go to higher voltage than half supply but that's just something you have to live with and in most cases will only be noticed as a 'pop' in the audio as the power is turned on.
Brian.