How does a car radiator cool the engine?

The basic rule of thermodynamics is that heat goes from where it's hot to where it's not. The coolant in a vehicle takes (draws or sinks) heat from the engine, and this heats the coolant. The hot coolant carries off the heat, and then flows through the radiator. Air passing the elements of the radiator, because it is cooler, takes (again, draws or sinks) heat from the radiator's fins, which transfer the heat of the coolant out. The hot coolant loses heat energy as it flows through the radiator, and heat from that coolant is passing through the metal radiator elements into the air flowing past. The radiator sets up a relatively large surface area to that passing air so that "a lot" of heat transfer can take place. There is what is called a thermal gradient across the elements in the radiator. Hot coolant inside and relatively cool air outside those elements set the ends of the gradient. Heat goes from where it is hot, the coolant, to where is it not, the air. If you guessed that there must also be a thermal gradient across the metal of the engine between the combustion chambers and the coolant chanels in the block and head, you'd be right. That thermal gradient drives heat out of the cylinder walls and head into the coolant in the coolant channels. It's just that simple. Remember that any time a temperature differetial exists, a thermal gradient exists. A thermal gradient is the "force" of heat transfer in the science of thermodynamics. Easy as pie.