You shouldnt be touching stuff while its running unless you have a specific purpose in mind, like checking the fans or using a multimeter to check voltages.
The voltages inside a computer are pretty low, nothing is over 12V and most things are around 1-3V so its pretty safe for you in there, but the bigger issue is you shorting two things together and hurting them. The heatsinks you see him touching in that video do not carry electricity so they are safe to touch though they may burn you.
When you are working on the system, you want to be grounded, you can do that by touching the case which lets the extra charge on your body go into the metal of the case and the PSU. This makes you the same potential level as everything else in the case, when you get a normal shock from building up static electricity its usually on the order of 10-50kV which is why you want to touch the case as it brings you down to 0V relative to the case.
If you are working around sensitive electronics that are low power and low voltage you want to be grounded so you can shock any of it and damage it. If you are working around high voltage you really dont want to provide a nice path to ground for all that energy, so you dont want to be grounded.