Why did it only tarnish on the edges? I would have thought the
inside of the band should have tarnished as well. And perhaps the
outside. So methinks I have no idea what happened.
You said merely a quick wipe with the cloth cleaned it up. So it
doesn't take much to remove. Simple friction with the finger kept the
inside clean, and friction just from whatever, clothing, etc, kept
the outside clean. The edges, sitting to the sides, and between the
fingers yet not quite in contact with them, were protected. So the
tarnish (likely just a reaction to what's in the air, or liquids the
finger came in contact with) built up where there was less friction
on a daily basis to keep it clean.
That's my guess at any rate.