On the Main Line in his usual erudite style collected a plethora of the early writings about the origin of the custom. I suspect that the only one of these sources that is generally well known is the explanation found in the Kuzari:
79. Al-Khazari: I should like to ask whether thou knowest the reason why Jews move to and fro when reading the Bible?
80. The Rabbi: It is said that it is done in order to arouse natural heat. My personal belief is that it stands in connexion with the subject under discussion. As it often happened that many persons read at the same time, it was possible that ten or more read from one volume. This is the reason why our books are so large. Each of them was obliged to bend down in his turn in order to read a passage, and to turn back again. This resulted in a continual bending and sitting up, the book lying on the ground. This was one reason. Then it became a habit through constant seeing, observing and imitating, which is in man's nature. Other people read each out of his own book, either bringing it near to his eyes, or, if he pleased, bending down to it without inconveniencing his neighbour. There was, therefore, no necessity of bending and sitting up...
My problem with this explanation is that it really doesn't make much sense. Anyone who has studied chavrutah, or with three people, knows that you don't sway. At the very least there is not "Fast Swaying". When people read from the same book, each one reads a large passage. Normally then you move the book around. Even assuming they were using "Big Books" they wouldn't sway. There is also a simple limit of how many people can sit around a book facing the right way up. You would have to assume people were swaying and reading upside down. Interestingly when I brought up this objection in Yeshivah, some of my Yemenite friends immediately claimed this was exactly what happened in Yemen. However, when questioned, none of them actually saw it happen. I'm skeptical by nature, and especially over some of the claims my Yemenite friends make over what happened in Yemen.
I'm not saying it isn't possible. Clearly it is, however my own experiance has shown that this explanation is not likely. I await the Youtube videos to prove me wrong.