The short answer: why do you need a rug pad depends on which of four jobs you actually want done, and they pull in different directions. Grip, so the rug stops travelling. Cushion, so it feels better and the pile lasts longer. Floor protection, so grit does not sand your finish from underneath. And airflow, so moisture does not sit trapped against the boards. A felt and natural rubber pad at a quarter inch does all four adequately, which is why it is the default answer. But if your floor is vinyl or LVP, rubber is the wrong choice and can stain it permanently, and if your rug already sits under heavy furniture, you may want pure felt instead. Get the material right first, then the thickness, then the size.
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