How does a domestic appliance "know" how much current to draw?
Electrical loads such as appliances have resistance. That, together with Voltage determines the current they draw (see Ohms Law). 

Above the rated current, the wires in your wall plug transformer overheat and burn (if a fuse is not present).

The current provided by a switch-mode power supply on the other hand is generally limited by its design.  Usually if you draw too much current, the voltage starts to sag.

Finally, if everyone unplugged their appliances at once, the voltage in the grid would want to go up, but the generation side will back off to keep it stable at the rated voltage.  I'm told stories about the 3pm Tea-Kettle hour in England which used to be a stressful time for the power authority as everyone in the country plugged in their kettles, the power authority folks needed to balance the demand by increasing power generation and keeping the voltage at the nominal rate.