guessing it keeps the controllers secured to the hand/thumb so it
doesen't fly away when you swing it around :P

If you've got a DS it explains in the manual.
Basically it's an alternative to the stylus.

You put your thumb in the loop at the end of the strap and position
the bit of plastic under your thumb. The plastic is used to touch the
screen instead of your thumb doing so directly.
I tried it once or twice but didn't find it very convenient.