A modern picnic cool box is used to keep food cool inside it during the day. However, when you place a hot stew inside the box and close the box, the stew carries on cooking slowly. Why does this happen?
What you call a "modern picnic cool box", I assume, is also known as a "cooler" in other parts of the world. This is generally a well-insulated, most often plastic, container. It could be hard-sided or made of soft material. The container and insulation is designed to maintain a consistent temperature. So, cold things remain colder longer, but also hot things remain hotter longer. It's the same principle at play in an insulated drink container (Thermos is a common brand). Also at play in your question is a phenomenon called "carry over cooking". Food retains heat and continues to cook, even after it is removed from the heat. Usually not a concern with a stew, which slowly cools when you are finished cooking. However, when you place your stew into a cooler, fairly soon after it is removed from the heat, it will take much longer to cool. Carry over cooking will be impacting your final product.
The sides of the cooler are insulated, which means they reduce the heat transfer through them. If you put something hot on one side, and something cold on the other, their temperatures will stay different longer than if the cooler were not there. So if you have something cold inside and hot air outside, the cooler keeps the stuff inside cold. If the stuff you have inside is hotter than the air outside, then the cooler is effectively keeping the air cool; it's keeping the stuff inside from heating up the colder air outside. The physics are the same whether it's keeping stuff outside from heating up the stuff inside, or keeping the stuff inside from heating up the stuff outside.