Using electrical potential energy $V=\frac{1}{4\pi \varepsilon_0} \frac{Q_1 Q_2}{r}$ , a particle further away from nucleus has lower magnitude of energy.
Using Coulomb's law, a particle further away from nucleus experiences weaker attraction, hence less energy is needed to maintain orbit$^\star$ around that e-shell compared to a electron shell closer to nucleus, hence the one closer to nucleus supposedly should have higher energy.
$^\star$I know in reality $e^-$ does not orbit around a atom, but its position exists as a probability density or radial probability function.
Best Answer
The potential energy stored in a two like charge system will increase with decrease in distance between them. While for a two unlike charge system, the potential energy decreases with decrease in distance (means potential energy gets liberated if they come close), accounting for increase in attraction.
In the equation, you provided, the potential energy in the nucleus-electron system is negative. This means the potential energy of the system is liberated and hence indicate attraction of the nucleus with the electron (this is how they attain stability).
Hence a system comprising of an electron far off from the nucleus will have high potential energy stored in it, indicating they have sufficient potential energy that can overcome the attractive forces (means the attractive forces between the electron and the nucleus is less). This means the potential energy liberated by an electron far from the nucleus is very less. Hence the outermost electrons are less stable.
For an electron very close to the nucleus, the potential energy is minimum, which means the system comprising of nucleus and a nearer electron liberates most of it's potential energy (so that the system will now have a lesser potential energy) to have an increased attractive force, which in turn corresponds to maximum stability.
So a large amount of energy is required to liberate an electron from an inner most shell rather than an electron from the outermost shell. This is why we say that the electron in the outermost shell has a higher (potential) energy than the inner most shells. So a less amount of energy is needed to liberate the electron from the outermost shell.