How does the Sun burn a crescent shaped scar in your eyes during a solar eclipse? I was listening to NPR this morning and a man talking about the solar eclipse said that you cannot look at even a sliver of the sun because it could still damage your eyes. He noted he's even had patients with a distinct crescent scar.

I've seen a couple of patients over the years where, you know, you've got very distinct crescent-shaped scars from looking at a solar eclipse 
http://www.npr.org/sections/health-shots/2017/08/18/543745409/be-smart-a-partial-eclipse-can-fry-your-naked-eyes

On the radio version he even, perhaps hyperbolically, noted he could almost see at what point they took off their glasses because of the size of the cresent scars.
How does this happen? I would understand if I were to hold my hand over part of my eye and look up at the sun the unshielded part would be the only paart of my eye to get damaged, but if part of the sun is obstructed I would think there is enough beams of light coming down to reach all of my eye. 
For example if there is a person standing to my right, and the last sliver of sun is the (from our perspective) left side. The sun would reach and damage the left side of my eyes, and the left side of their eye, but wouldn't affect the right side of my eyes which are between me and them? How are no beams of light reaching there?
 A: Whenever a light ray passes through a lens, it gets bent.  In fact, this happens whenever a light ray passes from one transparent medium into another;  you've probably noticed this effect before, in things like the "bent straw" illusion.  What makes a lens special is its shape.  Specifically, its shape means that all light rays emanating from one point and passing through the lens will all end up at another specific point on the other side of the lens.  (Technically, this is only true for what are called "real" images, but that's all we need for this discussion.)  If you need help visualizing this, try playing around with this Flash applet;  be sure to enable the "many rays" option.  Turning on the "2nd point" option can help you visualize what's going on.
Now, imagine that the pencil on the left is the Sun, the lens is the lens (and cornea) in your eye, and the image formed on the right is formed on the retina.  This is how your eye normally works.  In other words, rays emanating from different locations in space are focused on different locations in your retina.  As you are looking at your computer screen right now, there is a small, inverted image of the computer screen being formed on your retina.  And if you were to look directly at the sun, there would be a small, inverted image of the sun formed on your retina.  But because the light from the sun is so much more intense than the light from your computer screen, and because the sun emits a lot of UV radiation (which is particularly bad for cells), it can damage the light-sensitive cells on your retina.
You also say that

I would understand if I were to hold my hand over part of my eye and look up at the sun the unshielded part would be the only paart of my eye to get damaged, but if part of the sun is obstructed I would think there is enough beams of light coming down to reach all of my eye.

Playing around with that simulation should convince you that covering half of the lens does not cause half of the image to disappear.  It does block half of the rays that are forming the image;  but that just makes the image dimmer, rather than causing it to disappear.   So covering half of your eye would not protect that half of the eye;  it would just make the image of the Sun on your retina a bit dimmer. 
(In my intro physics labs, I always ask students to predict what would happen if we covered half of a lens that's forming an image, and then ask them to do it and see what happens.  About 90% of them get it wrong at first, which makes them try to figure out what's happening, and reinforces their learning.  It's great fun.)
