Why does drinking alcohol make you feel warm?

The assumptions of the question are not totally true. It’s not just drinking, but where you drink and what kind of clothes you have on that determine whether the body ends up warmer or colder.

What happens is that alcohol causes the blood vessels in the skin to dilate, so that more blood passes through, and more blood means more heat going through the skin.

So if you drink and are sitting in the living room with a sweater on, for example, that heat due to vasodilation will be retained, and you will feel warm.

But if you go outside without enough clothing, all that heat will dissipate. Such heat loss poses a serious risk of death from hypothermia for people drinking outside in cold weather.

Excessive drinking also impairs sensation and the judgment that would ordinarily tell a person that it is time to come in from the cold.