It sounds like something went wrong on the RAMPS or Ardiuno board which has fried one or both of them. With cheap clone Arduino boards the 5V regulator is common to go (that’s the small black regulator between the USB connection and the 5.5mm barrel jack). If that goes it tends to screw other components. I know this as that is what happened to me. The LCD controller board was also fried… so you may have to replace that too…?
I assume this is a 12V system, and the power supply is a 12V supply? What amp rating is it? Switching power supplies should only provide what current is needed - it shouldn’t push all its available current through. If that was the case then yes - something probably would go “pop”! From what I can see in your images it looks correct. Connecting the PSU to the RAMPS board via the green connection is correct, which the RAMPS board sat on top of the MEGA. The RAMPS board will power the Arduino via a power connection which will go through that 5V regulator which I mentioned before on the Arduino. You don’t need any motors or hardware connected just to get the firmware uploaded and the board communicating with the PC. I would leave the Octoprint for now - you need to get the printer working correctly via a PC first.
You may find you need a new LCD controller board. If if one of your stepper drivers burst into flames - well, assume that is gone too! If you turn off the 12V PSU and connect the Arduino to the PC with the USB conneciton, it should power up the printer, just not allow any of the 12V systems to work - so the LCD screen should also work too when you get the firmware installed.
There’s a lot of guides out there on how to set this up.