Why does a worm's skin need to be wet for oxygen to diffuse across it?
The quick answer: When the skin dries, the lipids in the cell membranes of the skin tissue undergo a phase transition which makes the membranes less permeable for oxygen.

Explanation: The lipids of the cell membrane can exist in different phase states. In the liquid disordered phase the lipids are relatively flexible and mobile, making this phase more oxygen permeable compared to the liquid ordered phase, in which the lipids are more rigidly packed.

The phase transition temperature of lipids increases upon dehydration (another reference), meaning that at the same ambient temperature, a dry lipid membrane is in the liquid ordered state and a wet lipid membrane is in the liquid disordered state. 

Therefore, a dry cell membrane is less oxygen permeable than a wet one.