Why does your temp gauge rise when you turn on your air conditioner?

The air conditioner both consumes power and dumps heat as it pumps heat from the car's interior to the exterior world.

Most cars have an air conditioner radiator placed forward of the engine cooling radiator, because the engine does not need to be cooled to as low a temperature as the AC fluid. As a result, the air passing the AC radiator is heated before it reaches the engine coolant radiator, and the engine is cooled a little less efficiently.

The need to generate a bit more power to run the AC is a much smaller contributor to this phenomenon.