I'll take a stab at an explanation, but no guarantee my thoughts will get typed as they should.
For starters, to move voltage up and down without complicated circuitry the voltage has to be AC to work with a transformer. This dates back to when Thomas Edison first wired up cities to run lights and motors in the late 1800s.
A linear power supply has a transformer to change the typical 120V AC line voltage to something lower (for a 12V model) but a little higher than needed for the designed output of the power supply. Then the lower AC voltage is rectified into a pulsating DC, filtered with capacitors and sent to an electronic linear voltage regulator that keeps the final voltage fixed even with slightly varying AC line input and voltage drop within the power supply components under high current use.
There is nothing in most linear power supplies that would create any RF noise in the process of taking 120V AC down to a lower DC voltage. There are also unregulated versions of a linear power supply that simply use a transformer, rectifiers and filter capacitors to provide a DC voltage, but that voltage will drop some as you draw more and more current from the unregulated supply. That might be ok for charging a battery but not for sensitive radio equipment.
At the 60Hz line frequency used in the US, the transformer must have a lot of inductance to work properly and the core of the transformers are made of steel and are large and heavy. If you could raise the AC line frequency, the physical size and weight of the transformer can be much smaller and at some point lighter weight materials other than steel can be used like ferrite. 400Hz line frequency is common for some military and aircraft power and 400Hz power supplies are tiny compared to the same equipment that runs on 60Hz due to the much smaller transformers needed at 400Hz.
Enter the switching power supply. For a typical consumer switching power supply that converts 120V AC to a lower DC voltage like 12V, the power supply first and simply rectifies the 120V AC line voltage to DC and that typically ends up around 170V DC after rectifying and filtering.
But we need AC to use a transformer to change the voltage around in an efficient way, so the DC is made into a crude form of AC using a chopper circuit that might operate at a few hundred Hz or a few thousand Hz or even up to several MHz. This chopper circuit and the resulting square wave AC is full of RF interference that is hard to get rid of.
Now the dirty square wave AC is fed to an appropriate size transformer for this new high frequency, which can now be the size of a golf ball or smaller (instead of a grapefruit or cantaloupe for 60Hz), then more traditional voltage regulators can be used to keep the end voltage constant. Filter capacitors can also be much smaller at the high switching frequencies due to less capacitance needed to to the job. The end result is a power supply that can be 1/10th the size and weight of a linear power supply that does the same job.
Switching power supplies will have RF filtering at both the input and output to try and keep RF interference generated in the switching process from leaking back into the AC line or the DC output, but most are not filtered enough to work with sensitive radio equipment. When shopping for a power supply for radio use, you have to make sure its designed and rated for such use and it will have adequate filtering to keep RF interference to a low level.
Someone with some technical expertise should explain the characteristics of switching/non-switching power supplies in a pinned post so that the less savvy don't buy something that's not going to work. Switching power supplies are marvelous things. They make it possible to create cellphone battery chargers smaller than an acorn, and are produced in the billions, but they contain oscillator circuits that kick out lots of RFI due to the rapidly switching transistors that control the output voltage. Noise suppression is not a big part of the design brief, since these things are built for literally pennies.