The nature of the soul is a deep and complicated topic, but it can be summarized as our spiritual life — that is, who we are in terms of our thoughts and affections. And that, generally, is the internal sense when the Bible talks of a soul. You could be missing limbs, disfigured in the body due to illness or accident, paralyzed, but you would still be a complete human being. That's because the soul is the real you, and the soul has to do with internal things, not the state of your body.

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.