What determines the voltage requirements of an IC?
However when the micro-controller are being manufactured what determines their operational voltage?

All ICs are fabricated in a certain manufacturing process. That process is the recipe to make the transistors and other components.
That process together with the design of the transistors (large or small) also determines what the maximum allowed voltages are that the transistors can handle.
That process also determines the maximum speed at which an IC can operate. It also determines the power consumption of that IC. Note that the design (the circuits in an IC) also plays a big role here, fast and complex circuits generally require more power than slow, simpel circuits (assuming both are made using the same technology).
If you would use the transistors above that maximum voltage the transistors can suffer damage. The lowest usable voltage is also process dependent, for CMOS technology the threshold voltage is an important factor there. That threshold voltage is also process dependent.

For example, an atmega chip requires 5V but why is it 5V, why couldn't it be designed to operate at low-voltages like 3.3V or lower

It can, look at the datasheet of an ATmega328p and see that it can operate down to a supply voltage of 1.8 V! It can go even lower but then you're going below the operating voltage at which the manufacturer guarantees proper operation. So you can use the ATmega328 at 1.5 V but if it shows weird behavior, it is your own fault for using such a low supply voltage.