From my understanding after reading the research:
+ A big part of anxiety is a neutransmitter called GABA which is tied to stress control (GABA is a cortisol antagonist). Less GABA is typically associated with more anxiety and stress.
+ Alcohol acts like a fake version of GABA, giving you temporary euphoria and calmness when you drink.
+ In being a faux GABA, it fills all your GABA receptors and your body stops producing more GABA, thinking it's already full up.
+ Having less GABA and no longer consuming alcohol, you feel anxious.
+ Eventually your body produces normal amounts of GABA again (in healthy people)
What initiates the drinking can of course be chasing that GABA-ergic feeling to begin with, and this may be part of the mechanism of alcohol abuse. Ultimately this can be a spiral of behavior where having an anxious day makes you want to drink to "calm down", which in turn makes the anxiety worse the next day, on and on.