The silver nitrate is in solution and the metallic copper will dissolve to form copper nitrate; as it does, it will precipitate the silver in solution as metallic silver.
In other words, copper replaces the silver in the solution, and the copper that was not initially in the solution takes the place of the displaced silver.
Below is the provided chemical reaction involved in the process:
2AgNO3(aq) + Cu(s) \(→\) Cu(NO3)2(aq) + 2Ag(s)
(Silver Nitrate) (Copper) (Copper nitrate) (Silver)