Endotoxins are produced from the death of bacteria particularly. The implication is that fat is antibiotic in the gut. These endotoxins stimulate an immune response. The link I posted illustrates a mycoplasma (chlamydophila pneumonia) infection as the primary cause of MS. If the immune system recognizes the infected cells, they may kill them when fighting the infection. So, boost immune response, kill infected cells. On top of that, mycoplasma infect immune cells so that even though they go to the infection site to fight the infection, they re-infect the same site simultaneously. So, boost immune response, kill infected cells, re-infect healthy cells. The end result is a progressive destruction of tissues.
Conversely, reducing fat intake inhibits endotoxins release, in turn inhibits immune response, infected cells survive, healthy cells are not re-infected. The end result here is survival of the host, relatively speaking of course.
In this explanation above, we can see why MS is characterized as an immune disorder, but in fact it's an infection, the immune response is normal - not disordered - but the tissues are destroyed and re-infected constantly by the immune cells themselves which are infected by the same mycoplasma. This is clearly demonstrated by the link I posted about MS, where antibacterial treatment - no diet manipulation - has the potential to cure MS outright.
One factor could be vitamin A, which is essential for the immune system, especially for an active on-going infection, where vitamin A gets depleted quickly. A low fat intake would reduce vitamin A intake, therefore lead to vitamin A depletion even more quickly, and eventually to immune suppression. Now we have two ways to suppress the immune response, the killing of infected cells, and the re-infection of healthy cells. But vitamin A depletion comes at a significant price, since it's involved in red blood cells regeneration, meaning that the host gradually loses strength, energy, vitality, endurance, etc.
I doubt anything said about those experiments. I suspect they never compared MS to a healthy person when determining vitality. They likely only compared sick-vs-sick, so a tiny bit of vitality would look much more important than if compared to the healthy where vitality was not restrained by the disease itself. You know how it would go, "Oh, you look good today, for somebody with MS". Another thing is the psychological effect of MS so that any self-reporting will be almost completely unreliable. The mental state here is "don't care 'bout nothin".
Bear in mind, the explanation above is my understanding, I don't actually know if any of it is true, but it makes sense to me.