How to make sense out of Potential Energy?

This is very close to a philosophical question, since certainly Potential Energy is neither observable nor directly measurable. I have on occasion had to listen to non-Physicists insist that PE is entirely a "fudge factor" to maintain the law of conservation of total energy.

However, since every experiment to date shows that the PE models lead to subsequent action (motion, interactions, etc.) which perfectly match the mathematically calculated potential energy due to particle binding, position in a field (gravitational, electromagnetic, e.g.), there's no reason to deny the reality of potential energy's existence.

Work is required to move a positively charged object from position 1 to position 2 against an external electric field. The field can do the same work to move the object back to its original position. The object had potential energy in position 2 (relative to position 1). (This is analogous to lifting a mass in a gravitational field.) One might say that (in both cases) this energy is embodied in the changes which occur in the resultant field. In the case of charging a parallel plate capacitor, one can calculate the work required to charge the capacitor. If this work is divided by the volume of the electric field which is created between the plates of the capacitor, one gets an expression for the energy density in the field.