How to externally power HC-05 module with Arduino Pro Mini?
The HC-05 (the PCB that is mounted on the PCB of the module) itself runs on 3.3V. Many microcontrollers run on 5V, so people are selling the HC-05 mounted on a PCB with extra components, for example a voltage regulator from 5V to 3.3V.

Voltage regulators have a minimum input voltage for which they can hold the output voltage at the specified value. Refer to this question for more information about this. So supplying the module with 3.3V instead of 5V will result in a supply voltage for the HC-05, that is significantly lower than 3.3V, so it most likely won't work correctly.

If you want to use the HC-05 module as it is, you have to provide 5V supply voltage from another source. In this case be sure to connect the grounds of both power supplies, so that they have a common ground.

If you don't want to use an extra power supply, than you may bridge the voltage regulator on your modules PCB and provide the 3.3V as supply voltage, that are used by the Arduino. Doing so would not be an easy task for a beginner, since you would have to solder small SMD components. Also the outlay of the modules PCBs are very different (depending on manufacturer). Note that in this case you should never ever put 5V on the Vcc pin of the module again, since it is then directly passed on to the HC-05 chip, which would be destroyed from the 5V supply.