What is A in the following equation
$A $ is the cross-sectional area in solution. The  "bulk" is the relatively homogeneous "mess" of charged particles and solvent -- $A $ is not the area of the electrode.

What this equation tells you in terms of $A $, is that, the larger the area through which some number of charges are passing in a given moment, the greater the possible current, because more area means more room for more charges that can pass through in a moment. If you squeeze the area you start to make a bottleneck wherein less charged particles can pass in a given time. (This is why copper wires with larger diameter (or, smaller US gauge) have higher current capacity).

This makes sense, right? Current is measured as charge per unit time. The implication there, is that it is motion of charge through some point (1D) or cross sectional area (2D, $A $ here) per unit time.