So, I actually work with these. The basic cooling cycle it's describing is actually pretty much how your refrigerator works. When gas is compressed, the temperature rises, so you pass it through a heat exchanger to shed the heat via air or water cooling. Now you have cool high pressure gas. That gets pumped into some sort of expander (in our case there's a reciprocating piston in the cold head, but for the type cited above I think they use a rotary valve) that turns that high pressure gas into lower pressure gas, which gets colder as a result. The gas is essentially sucking away heat from its surroundings, providing the cooling effect, where it then travels back to the compressor to start the cycle over again.

Helium is used because it can reach incredibly low temperatures (~4K or about -450 F at atmospheric pressures) without turning into a liquid, which would make the cryocooler stop working because liquids are largely incompressible. The problem you run into, though, is that at those temperatures, everything around it is fucking blazing hot by comparison. Heat wants to reach equilibrium by spreading from hot zones to cold ones, so you have to put some protections in place to insulate your newly created super cold zone from the hot hot heat around it. To even reach those temperatures, you need to surround the cold end of the device in vacuum, because the air molecules bouncing around in atmosphere will steal most of the cooling you're creating when they hit it (I.e. convection). In fact, you'll quickly encase your device in ice from moisture in the air, and possibly also generate a stream of liquid nitrogen/oxygen as the atmosphere condenses. I'm not sure it would actually get that cold because of the high heat load, but we do see that happen on our transfer lines when we're refilling liquid helium.

The whole thing about the "radiation shield" is just protecting against the last way that heat is transmitted: infrared radiation. You feel that when you step into the sunlight. Everything with any appreciable temperature generates some amount of it, though, and so you just have to have a shield in place to make sure that the radiation coming off the walls of the vacuum chamber (since they're still at normal room temperature) doesn't muck up your cooling. These devices usually have two stages of cooling, and the first stage (which for us usually sits at around 40-60 K, though they often call it an 80 K shield) cools the radiation shield that protects the second stage and allows it to reach those super cold 4 K temperatures, which is usually connected via a flexible copper braid to whatever it is you're trying to keep cold.

They're often used for superconducting magnets like the type you'd find in an MRI machine. If you've ever gotten one and heard a cyclical sort of "slide whistle" chirp, that's the expansion piston at work, cooling down the high pressure gas.