What is the benefit of high surface area electrodes?
A current is simply the rate of flow charge (and hence the rate of flow of electrons); the more electrons flowing, the higher the current.

Anions migrate across the potential gradient between electrodes, carrying the electrons to the anode and the anions are then oxidised.  The oxidised species (often uncharged) must then diffuse across a concentration gradient to move away from the electrode (via Fick's laws).  Until the oxidised species moves away from the electrode surface, they are blocking the surface for more anions to migrate in, and hence the current is eventually limited when the rate of migration of anions is equal to the rate of diffusion of the oxidised species.  This exchange happens at a fixed rate per unit area at the surface of the electrode.

When we use a larger surface area electrode, there is more area for this exchange to happen, therefore more species involved in the exchange, therefore more electrons can be deposited in a given timeframe, hence permitting a higher current to flow.