Calculating the electrode potential from thermochemical data
As stated above by @orthocresol, these values of $\Delta H$ alone are not sufficient to find the electrode potential. To do so, you would need $\Delta G$ for each reaction, so you would either need to find a table of $\Delta G$ for these reactions, or you would need a table of $\Delta S$ along with the table given.   

But your question is why do we care about those particular values to begin with. The reason for this is that those reactions summed together give the reduction reaction for that metal. For example, I will look at the reactions of iron. 

The reduction reaction for iron: $$\ce{Fe(s) \to Fe^2+(aq) + 2e^-}$$

All the other reactions in the order they are listed in the original post:
\begin{align}
\ce{Fe(s) &amp;\to Fe(g)} \\
\ce{Fe(g) &amp;\to Fe^+(g) + e^-} \\
\ce{Fe^+(g) &amp;\to Fe^2+(g) + e^-} \\
\ce{Fe^2+(g) &amp;\to Fe^2+(aq)}
\end{align}

If you add together these reactions, you should obtain the reduction reaction. Consequently, by Hess's law for Gibb's free energy, the sum of the $\Delta G$'s for these reactions will give you the $ΔG$ for the reduction reaction. 

Then, it is just a matter of converting the $\Delta G$ to a reduction potential using the equation given in the original question:$$\Delta G^{\circ}=-nFE^\circ$$ where $n$ is the moles of electrons, $F$ is Faraday's constant, and $E^\circ$ is the standard reduction potential.