I'm going to use the Australian terms in my explanation to match the video. Here's the translation if you need it:
- Aus -> US
- Earth -> Ground
- Active -> Hot
There is voltage on the chassis because there is a Y Capacitor between either or both of the active and neutral lines to chassis. The chassis is normally connected to earth but on an unearthed outlet this connection doesn't exist. If it did then the chassis potential would be the same as earth. Without it the chassis can float to a higher potential than earth and therefore give a zap when you touch the chassis and your body forms the connection to earth.
While it is common practice to use Y-Capacitors, if there is enough voltage on the chassis to cause a zap when the supply is unearthed, then there is probably going to be enough current flowing in the earth when the supply is earthed to trip the RCD.
So perhaps your confusion is warranted - no electrical appliance should use earth as a current path, so it ought to raise some eyebrows if a voltage appears on that earth connection. However, Y-Capacitors are often part of a valid design, and they form a conductive path between the incoming voltage and the earth connection. The designer's job is to make sure their conductivity is low enough to be imperceptible.