Another explanation :
Hot water goes in the larger tank to increase the efficiency of the radiator. Heat transfer is dependent on temperature difference, thus, the most efficient part of the radiator is where the temp difference is greatest. So by making the most surface area available when the water is hottest, you get rid of more heat than having the input and output the same size. As the water cools down heat transfer drops and this becomes a problem when the heat transferrate drops to a point where it cant remove enough heat from the water before the water leaves the radiator.
