"To feel the wetness" means to experience or sense the sensation or touch of something being wet, usually due to contact with water or another liquid. Full definition
«Children don't feel the wetness in these disposable diapers, so they don't realize what's happening when they go,» Parpia said. (abcnews.go.com)
We really think being able to feel the wetness in these encouraged her to actually get up at night to potty instead of just using her diaper. (diaperjunction.com)
The added bonus, I found, was that she never lost the sensation of feeling wetness when she peed. (diaperjunction.com)