Why is electrostatic potential constant throughout the volume of the conductor and has the same value (as inside) on its surface?

Hint: Being a conductor, electric charges can flow freely through it. Therefore, any difference in potential will be immediately removed by a flow of charges.

Formula Used: The formulae used in the solution are given here.
$E = - \dfrac{{dV}}{{dr}} = 0$ where $E$ is the electrostatic potential, $V$ is the potential difference between two points on a conductor and $r$ is the distance between the two points.

Complete Step by Step Solution: A conductor has free electrons. As long as electric field is not zero, the free charge carriers would experience force and drift. When there is no current inside or on the surface of the conductor, the free charges have distributed themselves so the electric field is zero everywhere inside the conductor. Electrostatic field is zero inside a conductor.
As inside the conductor the electric field is zero, so no work is done against the electric field to bring a charge particle from one point to another.
$E = - \dfrac{{dV}}{{dr}} = 0$ where $E$ is the electrostatic potential, $V$ is the potential difference between two points on a conductor and $r$ is the distance between the two points.
Since $E = 0$, therefore the potential $V$ inside the surface is constant.
Now, we know that, electrostatic potential
Because there is no potential difference between any two points inside the conductor, the electrostatic potential is constant throughout the volume of the conductor.
$\Delta V = {V_1} - {V_2} = 0$
$ \Rightarrow {V_1} = {V_2}$
This proves that the potential at every point is same inside the conductor

Note: A neutral conductor has equal amounts of positive and negative charges in every small volume or surface element.
When the conductor is charged, the excess charge can reside only on the surface in the static situation. This follows from Gauss's law. This means there is no net charge at any point inside the conductor, and any excess charge must reside at the surface.