"Het oog kijkt naar mij."
Translation:The eye is looking at me.
Well, 'seeing' (zien) has to do with the ability to see, it's kind of passive: unless you're blind, as soon as you open your eyes and it's not too dark, you see everything around you, whether you want it or not. There's no volition involved.
Now, 'looking' (kijken) implies some sort of intention: you direct your eyes to a particular person/object/whatever in order to take a look ar them. There's volition/intentionality involved, you kind of do it on purpose.
Hope this helps.