In my understanding of the explanations I received:
As many Buddhist concepts, "seeing things as they are" is a pointer to something happening in real life, but not necessarily in a sense we assume. Its meaning is more practical than strictly technical.
Most of the time we are "in our heads". We have so many preconceptions, thoughts, and inner chatter going on, that we can barely see what's going on in the real world. Instead, our vision is limited by our thoughts and preconceptions to such a degree that we barely notice things outside of our frame of reference. We tend to ignore them for two reasons: either
A) in our frame of mind they are not cohesive phenomena, they seem illogical (because we don't understand their inner logic) - and so we tend to ignore them as noise, or
B) they go against our emotional attachment to some sort of theory or ideology, and so we semi-subconsciously dismiss them as fake / illusory / insignificant.
When we learn to suspend our judgement and lay aside our attachments and preconceptions (this is what Mahayana's realization of Emptiness entails in practice), and stop our never-ending inner-monologue, and lose solidity of this heap of overgeneralizations we call "ego" -- then and only then we start seeing things "as they are".
In my experience with it, this vision liberated from preconceptions can be allegorically called "quantum" or "multidimensional". Because we are not limited to one theory and one interpretation of what's going on, we see them all, or rather, we directly see the entire "space of interpretations". There is an element of ambiguity, but it's not regular ambiguity which is hazy vision - instead it is a very clear kind of ambiguity.
It's a very practical, very direct kind of vision. We are simply not in our heads, are eyes are open and we really look around, and see things we have not seen before.
This is the critical point of the explanation.
And then on top of that, as we develop prajna, we begin to "see" "hidden connections" between things. These are not as much some magic properties as abstract relationships of (latent) influence and causation.
Combination of this "quantum" vision and seeing the latent influences is called "seeing things as they are" - in my understanding.