Let's start from the basic. In an aqueous solution of Potassium Chloride, there are four types of ions: (Cations) Hydrogen ions and Potassium ions & (Anions) Hydroxide ions and Chloride ions.
When a concentrated aqueous solution of Potassium Chloride is electrolysed, Hydrogen ions are reduced to Hydrogen gas on the cathode whereas Chloride ions are oxidised to Chlorine gas on the anode. Therefore, as the electrolysis proceeds, the concentration of Hydrogen ions and Chloride ions in the solution decrease, thereby leaving behind Potassium and Hydroxide ions. As you may already know that Potassium Hydroxide is an alkali, the solution therefore turns alkaline.
Well, the more acidic a substance is, the more hydrogen it contains... the more alkaline a substance is, the less hydrogen it contains. So I think it has something to do with the fact that that the electrolyte is losing hydrogen atoms, possibly because they turn to +1 ions and react with the potassium which collects at the bottom of the container, leaving the electrolyte more alkaline than before.