Why does an Exhaust Gas Temperature (EGT) gauge require a power source?
The thermocouple voltage is a DC millivoltage. Presuming you’re talking about an analog gauge, there simply isn’t enough potential to “drive the pointer.” As such, that small voltage is usually amplified. The amplification requires that a larger power supply be used. 

For a digital gauge, that millivoltage is capable of being read by a microprocessor, but the rhetorical question of “What powers the microprocessor?” Should be enough to resolve the rest of the question.