Why does the eye pain so much when it is dry?

Well, here's the answer:

The transparency of the cornea is due to the fact that it contains hardly any cells and no blood vessels.  However, blood vessels can creep in from around it, if it is constantly irritated or infected, which can interfere with vision. On the other hand, the cornea contains the highest concentration of nerve fibers of any body structure, making it extremely sensitive to pain.  The nerve fibers enter on the margins of the cornea and radiate toward the center.  These fibers are associated with numerous pain receptors that have a very low threshold.  Cold receptors also are abundant in the cornea, although heat and touch receptors seem to be lacking. [Source]

– and from another source, this info

The cornea is one of the most sensitive tissues of the body, as it is densely innervated with sensory nerve fibres via the ophthalmic divisionof the trigeminal nerve by way of 70–80 long ciliary nerves and short ciliary nerves. The ciliary nerves run under the endothelium and exit the eye through holes in the sclera apart from the optic nerve (which transmits only optic signals).^[8]