The "King of Rock" style is a style used within hip-hop, where the rappers share lines within a verse or words within a line. I have chosen to name it after Run-DMC's 1985 song "King of Rock" because it is a style heavily associated with Run-DMC, and this song is one of the best examples. There are probably earlier examples of the style, however.
While duos and groups are common in hip-hop, the usual technique that they use is to alternate verses on a song. The connection is that the rappers are reciting rhymes around a common theme. Of course, since many hip-hop songs are free association, the connection might be tenuous. In fact, because the rappers aren't interacting with each other, they can phone in their performances, literally or figuratively. The "King of Rock" style instead, has the rappers interacting with each other, producing a heightened emotional energy as they rush to complete each other's sentences. It is an effect that I like, but which I don't know is used very much on modern hip-hop recordings.
I don't know why it is not used much. I have heard examples of the style used both as a high energy, shouted party rap (such as on "King of Rock" itself, and in a softer, lyrical form (on Hieroglyphics "Mics of the Roundtable"), so it is not just a matter of hip-hop becoming more sophisticated. I would actually say that the reason for its decline may just be laziness, since getting it to sound right involves more careful technical and artistic collaboration between the artists, producer and engineers than simply recording verses separately and laying them down on a track. I also think it is interesting that some of the best examples I know come from Run-DMC, Hieroglyphics and the Wu-Tang Clan, groups whose connection goes back to childhood, and who have an intuitive ability to act together. In other words, as a style, it might not be able to be learned, but simple grown into.