What's interesting to me is that he is quite particular about making sure that a bracelet will be waiting for him when he's done with an activity during which he is not allowed access. This morning he carefully placed a bracelet in the very middle of the back seat of the car, knowing that it would be there for him when I picked him up this afternoon after school. It was the first thing he reached for when he got in the car hours later!
What does this do for him? I can't give you a definitive answer, but I will make an educated guess. It keeps his internal busy-ness occupied.
When self-advocates (those with autism who can communicate in some manner, tell us what they want/need, and "translate" for us NTs - NeuroTypicals) discuss repetitive behaviors, many times they will say that it calms them when overwhelmed. I can't say that it calms him. In fact, he can get quite taken away with flipping the bracelet and vocalizing. But I do believe that in circumstances that can be overstimulating (shopping malls/groceries, restaurants, crowds, etc), the bracelet gives him something to distract him from the overwhelming buzz that is our noisy, crowded world.