It does not move out of the nitrate ion where it has an oxidation state of plus five. The other two have to change because they both are elements with a zero oxidation state on one side and in compound on the other. Silver goes from plus one to zero and Copper goes from zero to plus two. Think of this on a number line. The copper is oxidized because its oxidation number goes up from zero to plus two.
Half reactions are either an oxidation or a reduction.
Only the species of atom that is involved in a change is in a half reaction. In the above reaction, silver goes from plus one to zero oxidation state, but to account for everything, the electrons must be placed into the half reaction. The other half reaction is that of copper. This time the material is oxidized and the electrons must appear on the product side. We must double the silver half reaction to cancel out the electrons from right to left.