The dish-detergent method is used when you turn green wood and want to go to your end form/product very fast, i.e. shortening the normally monthlong drying phase. I think the idea behind the whole thing is that the detergent reduces the surface tension of the water in the wood and replaces some cell water through osmosis. The person who invented this method is Ron Kent, a pretty famous turner in Hawai'i.
I have used it on some bowls, and it seems to work fine. The only thing that I found negative is that I had more bowls breaking apart and pieces flying through the air when I had a catch, what happens to me in bowl turning very often. Also, the wood turned much darker than anticipated, but my experience is mainly with Maple.