When surface area of electrodes is increased, why does the current in the galvanic cell increase?
Yeah, you're on the right track. In any battery reaction there is some rate-limiting step that determines the maximum power the battery can supply. Most of the possible rate-limiting steps involve the surface of the electrode.

As an example, imagine that the rate-limiting step is the diffusion of new ions from the electrolyte bulk to the cathode (or diffusion of ions away from the surface at the anode. The physics of diffusion in this case is governed by Nernst-Planck equation, where the quantity of interest is the ion flux (number of ions of unit area). However, the total current is determined by the total number of ions. To get from flux to total number of ions you have to integrate over the area of the electrode: if you double the area of the electrode you'll double the total number of ions (ie. current) if all else stays equal.

That's just one example that depends on the specifics of your cell, but many of the surface-related limited steps in a battery follow this type of behavior.