If you plug in a bridge rectifier directly to a power outlet, and touching one of the outputs, isn't it like touching live wire?
First of all, it is always good practice NOT to touch live power wires. In the case of the video you mentioned and, almost all appliances and power supplies, there will be an isolation transformer connecting the mains to the circuit you are touching.

For a single phase, there are live, neutral and earth wires, on the mains side. The neutral and earth wires will be connected together and, ideally, be at the same potential. If you touch the live wire, you will likely form a circuit to the earth and be shocked. If a 1:1 isolation transformer is in between, you'll just have live and neutral wires effectively with no direct connection to earth, and if you touch the live wire on the secondary side, it is less likely a connection to earth will be formed (it can still happen though so don't touch)

There is a good discussion of this here: https://www.soundonsound.com/sound-advice/crosstalk-readers-writes.