During any electrochemical reaction, oxidation happens at the anode and reduction happens at the cathode. In an electrolysis reaction, an external source of voltage is required to make the reaction proceed.
Since oxidation is happening at the anode, we see that we are losing electrons in a reaction such as
Fe(s) --> Fe2+(aq) + 2 e-
We are going from the solid iron to aqueous iron so the anode (i.e. the metal electrode) will be losing mass. We will decrease the mass of Fe(s) and increase the amount of Fe+(aq) ions.
For the cathode, we have a reduction reaction happening. For example
Zn2+(aq) + 2e- --> Zn(s)
Therefore the cathode is the solid zinc and we are forming more of the solid zinc from the zinc ions in solution gaining electrons (i.e. being reduced) and forming solid zinc so the mass of the cathode is increasing.