Step 1: Nature of reaction.
When a copper rod is dipped into AgNO$_3$ solution, a redox reaction takes place. Copper is more reactive than silver and displaces it from silver nitrate solution.
Step 2: Chemical reaction.
\[
Cu (s) + 2AgNO_3 (aq) \rightarrow Cu(NO_3)_2 (aq) + 2Ag (s)
\]
Step 3: Explanation of colour change.
- Cu(NO$_3$)$_2$ is formed in the solution, which contains Cu$^{2+}$ ions.
- Cu$^{2+}$ ions impart a blue colour to the solution.
- Silver metal (Ag) is deposited on the surface of the copper rod.
Conclusion:
The solution turns blue because Cu$^{2+}$ ions are formed in solution, and silver gets deposited on the copper rod.