Why does the copper anode dissolve? In such a setup, the copper anode is known to dissolve. My question is, why? Does it not receive electrons from the $ \ce{OH^-}$ ions. And if you say that those electrons just flow to the battery, let me provide the counterargument: the electrons flowing to the cathode do not prevent the copper cathode from dissolving because copper ions in solution accept the ions, so there is actually nothing stopping the cathode from dissolving as well. So, very simply, my question is this:  Why does the copper anode dissolve?
The anode is where oxidations happen (the cathode is where reduction takes place).

In the vicinity of your anode, you have the following compounds: $\ce{H2O}$, $\ce{Cu}$, $\ce{Cu^2+}$, $\ce{SO4^2-}$. If you check oxidation states, you will realise that oxygen is always $\mathrm{-II}$, hydrogen is $\mathrm{+I}$, sulphur is $\mathrm{+VI}$ and copper is present in both $\mathrm{\pm 0}$ and $\mathrm{+II}$. Thus, the only species available for oxidation are copper($0$) and oxygen($\mathrm{-II}$).

Checking standard potentials:
\begin{align}
\ce{O2 + 4 H+ + 4 e- &amp;&lt;=&gt; 2 H2O} \qquad &amp;E^\circ = \pu{+1.23 V}\tag{1}\\
\ce{Cu^2+ + 2 e- &amp;&lt;=&gt; Cu} \qquad &amp;E^\circ = \pu{+0.35 V}\tag{2}
\end{align}

It should be clear why copper is oxidised.

On the reduction side, hydrogen, copper and sulphate all could accept electrons. Copper is again the happiest of the three to accept electrons.