All these points, marked with red arrows, are connected together, physically, with metal and wires:
For example, the oscilloscope's "low-side" is the outer metalic shield on the female BNC connector at the front of the device, which is connected via the chassis and power cable of the oscilloscope directly to mains earth.
The probe's ground clip is connected directly to its own male BNC connector's shield, which when plugged into the oscilloscope will also be connected to mains earth via the oscilloscope's chassis.
The bench power supply's negative terminal may or may not be connected internally to mains earth. I'll assume it is in that picture, because I see no separate green earth terminal.
Let's say you build the following circuit, powered from the bench supply, and you wish to measure the voltage across the LED using the oscilloscope:
simulate this circuit – Schematic created using CircuitLab
You've been told that the voltage across a green LED is about 2V, right? You simply want to see this for yourself. This looks innocent enough, until you draw in the "hidden" earth connections:
simulate this circuit
If you look carefully, you'll see that you completely short-circuited the resistor. There might as well be a direct wire across it, because that's exactly what is happening if you connect you oscilloscope's "low-side" to the bottom of the LED.
Effectively we are applying the full 12V of the power supply directly across the LED, which will "release the magic smoke", and die a really horrible death. If you don't take great care, you can severely damage the oscilloscope too. Imagine the enormous current that would flow around the earth loop, via the oscilloscope, if you were to accidentally touch the probe's low-side clip directly to the power supply's positive terminal.
The "take-away" is that you really can't use a mains-powered oscilloscpe to measure voltages across "any old thing", in the way you can with a battery-powered multimeter. You must always be aware that the probe's low-side is permanently connected to mains earth, and if any part of your circuit under test is also connected to mains earth, you have the "potential" for disaster (pun intended).
Battery powered equipment has no "hidden" connections to mains earth, and therefore can't cause/suffer this kind of catastrophic destruction. There are of course other great ways to destroy perfectly good and expensive equipment.