How does the dual 5V and 3.3V output on the Uno R3 work exactly?
Pins of the UNO/Nano/Mini/Mega board are connected to GPIO pins of the MCU without any additional circuit. The ATmega MCU running at 5 V will sense 3.3 V as HIGH because of the threshold level to convert voltage to logic state.

ATmega will read the pin HIGH from 0.6 * Vcc, so at 5 V Vcc the threshold is 3 V.

Sources:


Schematics of Arduino Uno
Datasheet of ATmega328p (28.2 DC Characteristics)