Why is mass gained at the cathode?
As you have figured out, the zinc electrode is undergoing oxidation since zinc is a stronger reducing agent than copper.  

$$\ce{Zn -&gt; Zn^2+ + 2e-} $$  

Thus, the zinc electrode becomes thinner as shown in your image. It is oxidizing into soluble $\ce{Zn^2+}$ ions and thus its mass decreases.  

In the right solution, $\ce{Cu^2+}$ ions are being reduced to metallic copper which deposits on the copper cathode  

$$\ce{Cu^2+ + 2e^- -&gt; Cu} $$  

The $\ce{Cu^2+}$ ions are already there to begin with. The copper electrode isn't dissolving to form $\ce{Cu^2+}$ ions to begin with (the copper electrode does not react with the $\ce{Cu(NO3)2}$ solution), so there isn't any loss in mass of the copper electrode initially. Rather, during the operation of the cell, the reduction of copper(II) ions forms metallic copper, which causes the copper electrode to increase in mass and gets thicker, as seen in the image.