When we perform electrolysis of say an aqueous solution of silver nitrate with silver electrodes, silver ions, having a greater reduction potential than water, accept electrons and are deposited at the cathode. At the anode however, silver just leaves the electrode and enters into its solution.

The anode must just be a metal rod whose sea of electrons is much poorer than what it must be. Why does the silver get oxidized and leave the rod as ions? Water could get oxidized instead and that appears to be a more logical expectation.

Secondly, why does this not happen if we do the same reaction using platinum electrodes? What is preventing the metal from leaving the anode as an ion, moving to the cathode causing gradual thinning of one electrode and thickening of another?