The A/C condenser in front of the radiator is a big heat generator and removes heat from your vehicle and deposits the hot air in front of the radiator. The A/C pump also puts an additional load on your engine.
So the additional heat from the A/C condenser, load form the A/C compressor and the load from going up the hill generates a lot more heat that your cooling system can't get rid of so your car can overheat going up steep hills. Thus the sign warning to turn off A/C. It isn't as much of an issue on modern well maintained vehicles especially if you aren't towing anything.