Silver chloride is insoluble in concentrated hydrochloric acid, whereas potassium chloride is soluble.
The mixture of potassium and silver chlorides is treated with concentrated hydrochloric acid. Silver chloride remains undissolved, while potassium chloride dissolves. The solution containing potassium chloride is filtered, and the filtrate is evaporated to dryness. The solid residue is potassium chloride. The undissolved silver chloride can be filtered off and washed with water. It can then be dissolved in aqueous ammonia to form silver diammine complex, which can be further processed to obtain pure silver.
The reactions involved are as follows:
$$AgCl(s) + HCl(conc.) \longrightarrow No \space reaction$$
$$KCl(s) + HCl(conc.) \longrightarrow KCl(aq) + HCl(g)$$
