It's like using a magnifying glass to burn an ant on the pavement, but the sun is you, you're the ant, and the sun is using your eyeball as the magnifying lens
The sun is still the sun in this comparison. Its not like "you" is the source of light here. The back of your eye is the ant, and the lens of your eye is the magnifying glass.
Because the sheer amount of energy going directly on your retina will burn a hole through it. It'll also damage the photo receptor cells at the back of the eye.
On a related note, why is it that looking at the sky when there's a lot of light, if I use both eyes it hurts, but if I keep one closed I can see the sky just fine?
There is physical damage of heating the retina with focused light, and there is chemical damage of depleting the reactions that normally occur when light is translated into neural signals in the retina.
Wikipedia says that the heating from the focused light is minimal because the retina is surrounded with fluid, similarly to how a balloon filled, even partially, with water won't pop over a candle flame. However, the light itself is damaging, as far UV is ionizing radiation and can rip apart the molecules making up your cells.