It's common to say “I see” when we understand something. And indeed, “seeing” in the Bible represents grasping and understanding spiritual things. So it makes sense that the eyes, which allow us to see, represent the intellect, which allows us to understand spiritual things. This can also be used in the negative, of course; the Bible speaks of people having eyes and refusing to see, and the Lord when He was in the world advised plucking out an offending eye. These represent a refusal to acknowledge truth, or being misled by falsity.

4409. Because the sight of the eye corresponds to that of the understanding it also corresponds to truths, for everything seen by the understanding has a connection with what is true. It is also connected with what is good; that is to say, a person should not only be aware of what is good but should also feel an affection for it. Everything seen by one's external sight likewise has a connection with what is true and good, since these have to do with the way objects are shaped and so with their beauty and the delight this gives. Anyone who is clearheaded can see that every single thing in the natural creation has a connection with what is true and good, and from this can recognize that the whole natural creation is a theatre in which the Lord's kingdom is represented.