Copper is more reactive than silver ($E^{\circ}_{Cu^{2+}/Cu} = +0.34\text{ V}$ and $E^{\circ}_{Ag^+/Ag} = +0.80\text{ V}$). If a simple salt like $AgNO_3$ is used, Cu will spontaneously displace Ag from the solution via the redox reaction $Cu + 2Ag^+ \rightarrow Cu^{2+} + 2Ag$, leading to a loose and non-adherent deposit of silver. To get a smooth, uniform, and adherent coating of silver by electrolysis, the concentration of free $Ag^+$ ions must be kept very low. $K[Ag(CN)_2]$ is a highly stable complex salt that dissociates very little, providing a very low concentration of free $Ag^+$ ions. At this low concentration, copper cannot displace silver chemically, and deposition occurs only when an external electric current is passed.