I have a herniated disk in my lower back. why does it hurt so much when i cough or sneeze?

Stretch: The sneeze is causing a localized vibration/trauma which causes the spine to shift and impinge upon the nerve rootlets..

More pressure: Coughing, sneezing, or any other activity that causes one to strain or bear down increases the pressure in the abdomen, resulting in increased pressure inside of the disks of the spine. With a pinched nerve from a herniated disk, this causes the herniation to bulge a bit more, putting more pressure on the pinched nerve, resulting in increased pain.