For a long time scientists thought the disc didn't have a nerve supply and therefore couldn't feel pain. It's clear now that the inner part of the disc called the nucleus pulposus doesn't have nerve endings or a nerve supply.
But the outer covering called the anulus has two different nerve fibers and can therefore "feel" pain. It does so by transmitting pain messages from the nerves to the spinal cord and up to the brain.
Another way pain occurs is through chemical irritation. When the disc is damaged and has tiny cracks or fissures in the anulus, inflammatory cells spill out and irritate the nerve endings.
Tests show that some people have discs that are more chemically sensitive than others. And some people have discs that are more pressure sensitive than others. How or why this occurs is still a mystery.