EPE of a neutral object?

The paragraph from your book is too confusing. It is best to read more paragraphs preceding this one to have a thorough understanding.

In general, an ISOLATED neutral object should have zero potential, unless the object is under the action of an external electric field.

The electric potential of an atom is zero, because the potentials contributed by the +ve charges (i.e. protons) and -ve charges (electrons) cancel one another. But inside the atom, electrons are under the influence of electric field given by the +ve nucleus, these electrons possess -ve potential. Hence, work needs to be done in moving an electron from its orbit to infinity (which is the place where the potential is zero). Such work-done is the ionization energy of the atom.

After ionization, the neutral atom becomes a +ve ion. It can be viewed as a +ve charged object, and possesses a +ve potential. In other words, the work done in ionization raises the potential of the original neutral atom from zero to a certain +ve potential. The charged object (i.e. the ion) has an ability to attract outside electrons and becomes a neutral object (the neutral atom) again. This is the discharging process, and in which energy is given out. The potential of the object will fall from +ve to zero.