I'm not a biologist, so my answer will be from the physicist's point of view and understanding, but let's try:
Atmospheric plasma is a cocktail of reactive chemical species and also emits radiation. Some designs also heat the gas to pretty high temperatures, but this aspect is eliminated for the ones used in plasma medicine - so there should be no thermal effects.
The radiation comes in all spectral regions, but the most important here is the UV region (200-400 nm) - the energy of a single photon here is already sufficient to break chemical bonds, eventually killing bacteria (or healthy tissue, but our body can afford to lose some cells, the bacteria cannot). The same applies for the reactive species produced in the plasma jet (thsee are often referred to as reactive oxygen/nitrogen species - ROS, RNS or RONS) - again, they can attack a cell and cause damage, which eventually results in the cell death. Here, the biologists often distinguish between necrosis (you smash the cell like with a cannon ball, guts flying in every direction) and apoptosis (you persuade the cell to commit suicide itself - with plasma jets, you can achieve even this! - this is definitely cleaner and you avoid inflammation).
So far for the disinfection effects. By introducing the RONS, you create something called oxidative stress. Something similar happens to your body when you exercise hard and produce all the metabolites - this effect turns out to be good for your health in the end, because as the body starts to clean up the mess, it tends to end in better state than before (?? experts on living bodies, please help here, not sure anymore ??). By this effect you somehow support the healthy tissue to thrive whereas the bacteria die.
This pretty much sums up the things I have ever heard about the health effects of cold atmospheric plasma jets. A good start would be to start looking up papers with the words highlighted in bold.