I believe the situation in the video is such that a scope is used for debugging the unisolated mains side of a fly-back type switch mode power supply.
As no point in the unisolated side is at ground potential, you can't just poke a grounded scope there. I will explain both.
First, the reason why the mains side of the power supply is not grounded. Sure, in electrical wiring, ground and neutral wires are connected together, but in the power supply, neutral and live are full-wave rectified into a capacitor. So if you live in a 230V country, there will be 325V over the capacitor. But neither end of the capacitor is no longer at 0V ground level. During positive mains cycle, the negative leg of the capacitor is near 0V ground level and the positive leg is +325V, but during the negative mains cycle, the negative leg is at -325V and positive leg is near 0V.
The scope however is using the mains ground as 0V reference, and for safety reasons the metal case and probe ground lead are connected to mains ground.
So, with the scope grounded, you can (in theory) poke the probe tip to measure either leg of the capacitor, but positive leg will show a waveform between 0V and +325V while the negative leg will show a waveform between 0V and -325V.
But to show what is the voltage waveform over the capacitor positive leg in reference to capacitor negative leg, you would have to do one of a few things. Either get a suitable differential probe, or use two scope channels to make differential measurements, or the dangerous option, lift the scope ground and connect the probe ground lead to negative side of the capacitor.
The scope mains ground must be cut or otherwise there is a short circuit and tens of amps of mains current will flow through the scope ground clip during the negative mains cycle and your mains circuit breaker or fuse trips, likely with a loud bang, sparks and smoke, it may even damage the probe leads or the scope.
So when scope mains ground is cut, the scope is said to float, and then you can connect the probe tip and ground to the capacitor.
But as the probe ground is also connected to the scope metal case, it means, every exposed metal part on the scope and connected to the scope will now be live with rectified mains, wiggling between 0V and -325V, not safe to touch.
So if scope has other probes or anything else connected that are now floating at between -325V and 0V they will be deadly too. Or trip the breaker if you make a connection to grounded device.
But if you float the scope ground, the scope itself can be dangerous now, because it is missing a protection from the case becoming live due to an internal fault, not because you deliberately made it float to make it live.
Also the scope mains inlet likely has a filter which connects both live and neutral to ground through capacitors. It means that if you disconnect the mains ground, the scope ground, probe ground, and metal chassis will be floating at 115VAC because of the filter capacitors forming a capacitive divider. That may not be an issue, but again poking the scope tip or ground to a sensitive circuit might blow it up by discharging a capacitor with instantaneous voltage max +/- 162V through your circuit.