electromagnetism - What is potential energy truly?

I have a problematic question for which I have been unable to attain a satisfactory answer. What is potential energy truly?

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I have read about how potential energy can be seen as the "highering" of an object within a field, be it gravitational, electromagnetic, etc.

In a way, an object is lifted to a higher position, and when it eventually falls down again energy is released back.

The object does not even have to be "lifted" beforehand, as with the case of asteroids.

However, since energy cannot be destroyed, it must somehow be stored, yet how does energy truly store itself within a field? It seems hard for me to grasp that energy can physically be stored within the positioning of an object?

I can't really get my head around how a field conserves energy within it. Kinetic energy on the other hand, can be, for me, intuitively understood, as the object really is "moving" and energy manifests itself through motion.

Can anyone elaborate on this, perhaps give reasoning as to why a field can in fact "store" energy? Please point out any fallacious or naive views I seem to hold.