How do electrostatic analysers act as kinetic energy analysers?
The point is that all ions produced in the ion source will not have a kinetic energy outside of the acceleration region equal to $q V$. When produced the ions will most probably be produced at (slightly) different positions in the acceleration field, with a small $\Delta V$ potential deviation from the average acceleration potential, and with a thermal kinetic energy distribution ($E_{kin,thermal}$).

In these conditions, the real kinetic energy of the ions as they exit the ion source will be slightly different:

$$E_{kin,real} = q V + q \Delta V + E_{kin,thermal}$$

and the purpose of the electrostatic analyser is to separate the ions depending on their actual kinetic energy and thus compensate for the distribution of kinetic energies of the ions exiting the ion source.