I'm not pro-rollkur, but here's a few bits: http://www.scienceofmotion.com/Rollkur.pdf
"In the mid fifties, Jack Licartreferred to the lowering of the neck as an “extension”. The words were in agreement with the author’s philosophy; the thought of a neck extending out of the shoulders and elongating eventually the back muscles."
Pretty much, many rollkur advocates believe that the lowering of the neck and curling stretches the back muscles, even though that isn't true. Some also believe that the greater 'mobility' of the horse's vertebral column eases the horse. This is also not true. Another point they make is that the lowering relaxes the neck muscles and makes it easier for the neck muscles, but inherently this is fault as the stress is then put on the lower neck muscles and ligaments, so yet again this is not true. I guess you can just cut out the "it's not true" part
From Chronicle HF, an article: Huff And Puff Training And Leadership: An Opinion On The FEI Rollkur Decision | The Chronicle of the Horse
From an FEI meeting in 2006:
"Two major conclusions were reached at this meeting, which the FEI formulated as following: "There was clearly no evidence that structural damage is created by this training exercise, when used in the right way by expert riders. However, the use of that technique by inexperienced people was a possible threat to the welfare of the horse. The role of top dressage riders as role models in the sport was underlined. Most of the participants agreed that the terminology “Rollkur” was not comprehensible and decided it would be better to use a term which could be understood by riders, trainers and the general public. After an extensive discussion, it was proposed that the draft wording might be ‘hyperflexion of the neck.’ "