The reason for the phenomena is that the reaction is a redox (reduction-oxidation) reaction. AgNO3 in the solution splits to form ions.
`AgNO_3 -> Ag^+ + NO_3^-`
Copper is reducing silver ions (Ag+) to silver and eventually turning the color of the solution into light blue (because of Cu2+ ions). The reason why silver stick in the copper wire is that there is an exchange in substance happening in the boundary of the metal wire and the solution. As the copper metal goes to the solution to form copper ions, silver ions sticks to the copper metal to form the silver metal.
The complete reaction for this process can be written as:
`2AgNO_3 (aq) + Cu (s) -> Cu(NO_3)_2 (aq) + 2Ag (s)`