When a parallel plate capacitor is connected through a cell, each plate of the capacitor receives a charge with the same magnitude, but with the opposite sign. Is it because of the battery or the area of the plates?
Suppose you have a simple circuit with a capacitor and a power supply:
You want to create a charge on the capacitor, so you turn on the PSU to add some extra electrons to the upper plate:
But the number of electrons in your circuit is constant. The power supply can't create or destroy electrons. All it can do is act like a pump to move electrons round the circuit. Specifically it can pump electrons from the bottom plate round the circuit to the top plate:
So for every electron that you add to the top plate you have to remove an electron from the bottom plate, and that means the negative charge on the top plate is necessarily always equal to the positive charge on the bottom plate.
The battery creates a potential difference between its terminals. Because, in a steady state situation, the total voltage through a closed circuit is zero, this will create, when the capacitor is charged, the same voltage over this component. Then, if the charges on the capacitor plates wouldn't be equal in magnitude (!), the electric field outside (and inside as well) the capacitor would not be zero. However, in steady state the electric field inside a (perfect) conductor, the wires in this case, is always zero.