The copper nitrate appears blue because the compound absorbs some of the color from light as it passes through the solution. If no colors get absorbed when light passes through a solution containing a certain compound then we see the solution as "clear," but if the compound in solution can absorb, say both "red" and "yellow", from light then what we see emerge from the solution is the color it couldn't absorb. In this case that is blue.
Transition metal compounds, especially of the first row metals, typically absorb strongly in the visible range of light. They do this because of the presence of splitting between the energy levels in the d-orbitals on the central (in this case Cu) atom. In the bonding to ligands such as NO3 and H2O to form an octahedral complex, typically the dz2 and dx2-y2 orbitals on the central atom are pushed up in energy slightly in comparison to the other d-orbitals creating a small gap in energy between the these two sets of d-orbitals. This gap is sometimes small enough that visible light can be absorbed to pass electrons back and froth between the two energy levels. What doesn't get absorbed comes out as our color.
A good way to start studying why this reaction occurs is to look into the concepts of oxidation and reduction in your text. If you look at a typical redox table and compare the standard redox potentials of copper (0.34V) to silver (0.80V) then you will find that silver has a stronger tendency to be reduced that copper. So in the reaction, silver is reduced to silver metal while copper is oxidized.