How are these plastic rings used on my Shimano disk brake mount adapter?
The stop rings go over the head of the bolt.  The grooves on the ring line up with grooves on the bolt head and encourage it to keep from rattling loose.  They are simply snapped or pushed over the head of the bolt once it is tightened in place.

I suspect that in some country, they may be a requirement for manufacturers.  However, I haven't ever actually used them on any of my rigs.  Some loctite and regular maintenance are what I tend to rely on.  But, if you have had a bolt rattle out on you before, perhaps they could make you sleep easier at night.