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Coolers keep things cold by wrapping them in an insulator.
An insulator in this sense is a material with many small, sealed air gaps(cells).
Polystyrene foam is a common insulator. If you look closely you can see many tiny holes, these are the air gaps.
If you put something cold in a cooler, the cool items will cool the air in the cooler, then the air in the polystyrene in the cooler wall.
The air in the polystyrene insulator remains cooler than than the outside air, as it is not free to move, and is less likely to come into contact with warm air.