Your kitchen fridge has a compressor, which turns a gas into a liquid, releasing heat (which you’ll feel at the back of the fridge). The liquid is then pumped round the inside walls of the fridge, where it draws heat from the contents; that turns it into a gas, which is pumped on to the compressor.
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A magnetic fridge works like this. Powdered gadolinium (with coarse grains for good heat transfer qualities) is put into a magnetic field. It heats up as the randomly ordered magnetic moments - the electrons with spin - are aligned, or “ordered”, by the field. The newly-acquired heat - a boost of between 2-5C, depending on the gadolinium’s original temperature - is removed by a circulating fluid, like a conventional fridge.

The magnetic field is removed and the gadolinium cools below its starting temperature as the electrons resume their previously disordered state. Heat from the system to be cooled - your fridge interior - can then be transferred to the now cooler metal. Then all you do is endlessly repeat. But unlike conventional fridges, which need very toxic chemicals, the only liquid needed for heat transfer is water, alcohol or, more likely, antifreeze.