I'll take a stab at this.
Diffusion is the movement of a substance down its concentration gradient. It's a passive process. In annelids, diffusion is taken advantage of for respiration. It's important to realize that annelids can't control the rate of diffusion. Oxygen is moving into the body (high concentration outside to low concentration inside) and carbon dioxide is moving out of the body (high concentration inside to low concentration outside). Both processes are needed to survive.
The slimy, moist skin is necessary because oxygen must somehow get from outside of the annelid to the blood vessels inside. The oxygen has to be dissolved into the moistness of the skin in order for diffusion to take place. It has to pass the outer layer of cells via diffusion to get to the closed respiratory system. This can't happen with dry skin.. there is nothing for the oxygen to be dissolved in, and diffusion isn't going to magically happen.
For the DAT, just remember that no moist skin = no diffusion = no respiration = death. Hope that helps.