Why is it cooler under an umbrella?

It is not always cooler under an umbrella. Just as an umbrella can shelter you from rain it can also shelter you from the direct rays coming from the sun. When it does so it creates a shadow and the shadow is cooler than the full sun. However just as an umbrella will not shelter you from rain when it is not raining, an umbrella does not provide a shadow at night or on overcast days, so you will not be cooler under an umbrella then.