Part of my evidence is how much better I felt when I started doing these 5-day FMD "dives". A chronic condition (chronic pain in one of my femurs) which I had consulted a doctor about without avail and which had plagued me for decades completely cleared up, making my experience of life significantly more pleasant. (The most likely cause of the chronic condition is damage to the "growth plate" in this bone through too much usage of it when I was a child.) There were other positive changes if my subjective experience of life is any indication (and according to my immunologist, it is a good indication—at least of immune-system function).
Our ancestors were required to do hours per day of physical exertion just to stay alive. Because of technological progress, it is possible for most people these days to survive with very little exertion, but we now know that living this way (doing very little exertion) is bad for a person with the result that most of us should regularly exercise to maintain our health. Similarly our ancestors were required to endure periods of "calorie deficit" (eating fewer calories than would be survivable if the deficit went on indefinitely). The body uses these periods to disassemble damaged proteins inside cells (autophagy) and to get rid of unnecessary and damaged cells (apoptosis). When a person never undergoes a calorie deficit, these processes do not happen or happen to a much lesser degree than is optimal.
IIUC the main experimental evidence for the benefits of FMD come from an earlier protocol called "calorie restriction" in which the number of calories is kept constant and is a little lower (25% or 30% IIRC) than the individual would prefer to eat or that most Westerners actually eat. This protocol was found to extend life in mice significantly. Then someone found that fasting can improve the same biomarkers that calorie restriction improves (and consequently can be expected to have the same beneficial effects on longevity). But fasting by eating zero calories for days has adverse effects (including sudden death), starting a search for a diet that has the same beneficial effects on the biomarkers without the adverse effects of zero-calorie fasting.
Valter Longo is the longevity research who devised the FMD.
I am _not_ an expert or a professional in this field.