Air on the Dirac Strings
Why MathOverflow recommends it
MathOverflow users link it as a visual demonstration of why SO(3) is not simply connected and why a 720-degree rotation is needed to undo a twist, the plate or belt trick.
Recommended by 2 people across 2 different months, 2009 to 2010.
What it is
A 2m21s computer animation from the Electronic Visualization Laboratory at UIC (1993), directed by Dana Plepys, showing how an electron's behavior under rotation can be understood through quantum mechanics. It includes a human demonstrating the plate (belt) trick and has commentary connected to Louis Kauffman.
What people said
Recommends Tim Porter · 2010-02 · MathOverflow · on “How does $\pi_1(SO(3))$ relate exactly to the waiters trick?”
There is a vide with commentary by Lou Kauffman: try [this video] or search Google using: Air on the Dirac Strings This is useful but is not the complete answer so I will leave this as a comment and not an answer.
Recommends David Lehavi · 2010-04 · MathOverflow · on “How to demonstrate $SO(3)$ is not simply connected?”
You have two different proofs (a.k.a. the cup proof and the belt proof) right here [this video] There are several different demonstrations of the second one (Harrison Brown linked to this video in a comment to an answer I submitted - the cup proof - in the question asking for "proofs w/o words").
Cites Harrison Brown · 2009-12 · MathOverflow · on “Proofs without words”
It doesn't feature Feynman, but here's a video of a human doing the plate trick (just after 1 minute in): [this video]