It is simply due to the voltage required to bias devices into linear conduction and their saturation voltages. Taking a BJT for example, it requires somewhere around 0.5 - 0.6V between base and emitter to pass enough bias current to work as an amplifier. If your supply was only 1V, more than half would be used just to make the transistor conduct and any negative (assuming NPN) input voltage would stop it conducting at all. If you wired the stage in darlington configuration, as is common in op-amps, you would need more bias voltage than the supply could provide. Add to the equation that you want the output pin to be able to swing over sufficient voltage range to be useful (VCEsat typically 0.2V and in reality you wouldn't want to reach saturation), it becomes clear that you simply run out of headroom to handle the signal as well.
So basically, although there are some op-amps for low supply voltage, most need enough supply to cater for the individual voltage dropped across each internal element while keeping those elements working in their linear region of conduction. If the supply is too low, the devices either cease to function or enter a non-linear region of their characteristics and distortion results.
Brian.