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.

491. The same things are meant by 'sons and daughters' in verses 4, 7, 10, 13, 16, 19, 26, 30, of this chapter, but the character of a Church determines that of its 'sons and daughters', or goods and truths. Here they are truths and goods that men perceived most clearly, for they have regard to the Most Ancient Church, the head and forefather of all other Churches that followed after it.