My best stab at explaining this paper. The authors have a protein in cell membranes that plays a very complicated role in inflammation and seems to be messed up in pwME. I am not up to explaining that part. In a nutshell, the protein, called SMLDP3B, lives mostly in cell membranes. Sometimes they are chopped out of the cell membrane by an enzyme, called PI-PCL, and this causes problems when the protein is released into the bloodstream (at least I think that's where it's being released. Maybe it's intracellular. At any rate, when this protein is chopped out of the cell membrane bad things ensue.
How to stop this? Well, with a drug that inhibits the chopping enzyme. For some random reason, that turns out to be this diabetes drug. Saxagliptin inactivates the enzyme in a bacteria, and it turns out it does so in human peripheral mononuclear blood cells as well. Note that this experiment was done on blood samples, not people! We don't know what effect turning off this enzyme will have on the immune system in a whole person. I'm not sure what dose you would use in a person instead of some cells in a petri dish either. And maybe the bigger problem: we don't seem to know exactly if SMLDP3B being messed up is a cause or effect of ME/CFS. Without knowing that, it's hard to know what the effect of the drug would be on the illness...