This is really just a discussion topic rather than a topic with a real answer. I don’t even know if Rare even had an explanation. Do the Kongs turn into the animal buddy and the animal buddy combines with them? Do the kongs disappear and are replaced with such and such animal buddy?
I've always thought that you control the actual animal buddy, so maybe the Kongs are hiding off-screen? If that rare sound effect of many giggling monkey noises plays at Toxic Tower, then the process probably tickles.