Electrolysis

In silver nitrate solution, silver nitrate dissociates to give silver ions and nitrate ions, while water is slightly ionized to give hydrogen ions and hydroxide ions.
AgNO3(aq) → Ag^+(aq) + NO3^-(aq)
H2O(l) = H^+(aq) + OH^-(aq)
(reversible)

During electrolysis :

Nitrate ions and hydroxide ions migrate to the anode. Hydroxide ions are preferentially discharged because hydroxide ions is oxidized more readily than nitrate ions.
4OH^-(aq) → 2H2O(l) + O2(g) + 4e^-

Silver ions and hydrogen ions migrate to the cathode. Silver ions are preferentially discharged because the concentration of silver ions is higher than that of hydrogen ions.
Ag^+(aq) + e^- → Ag(s)

Hydroxide ions are discharged and hydrogen ions are left unreacted. Therefore, the concentration of hydrogen ions becomes higher and higher, and the acidity of the solution increases.