How does viewing through two eyes give us a sense of depth? 
With ONE ray, your eye can never determine where the object is. Note that in your diagram, the eye can deduce the line along which the apparent image is, but to make a point, we need two lines! Who told the eye that the apparent object was directly above the real object? So there has to be a second ray of light. - Manishearth

I sort of get the final experimental result, but I don't understand 'how' or what exactly is going on here. Could it be explained in more details how precisely it is that viewing through two eyes give us an illusion of depth?
One of main confusion comes from the fact that I think that depth comes from prespective. For instance, have a look at this photo:



In the above photo whether I open only one or two eye, it seems so that that the feeling of depth is preserved. I suppose that it would still be if I was standing in the physical place where the picture is saying.
 A: The photo is, of course, flat. So your feeling of depth when gazing upon it comes from your experience in the 3D world. Indeed, when looking at a scene, say, 1 km away, the distance to your face is much larger than the separation of your eyes. Your eyes see essentially the same thing. Therefore, much of your experience of visual depth in the world comes from interpretation of a 2D image (which your brain is highly evolved to do). You have learned that farther things appear smaller, much as you’ve learned that, e.g., dropped things fall. Your brain uses this information without you being aware of its existence.
Stereoscopic vision is important for viewing scenes at a distance on the length scale of your body. In this case, the spread between your eyes significant, and your brain interprets the two slightly different images as  depth. First, try looking at an object a couple feet in front of you and alternately closing your left then your right eyes. This will give you a sense of the information your brain receives. Next, try closing one eye and trying to pick up small objects an arm’s length away. Is it more difficult to gauge depth than with two eyes?
