'Hit' can mean 'start to have an effect on' someone or something, often (but not always) in a bad or unfortunate way. The person had previously consumed some alcoholic drinks, and they finally started to have an effect on her, e.g. made her feel drunk, or made her unsteady on her feet, loquacious, abusive, etc.
It seems that in the song, the 'narrator' bought Lucille some drinks and they had the effect of making her talk about her unhappy marriage, and how she wanted a better life. Then, as everybody over a certain age will recall, her husband came and reproached her (in rhyme, it seems, which shows a certain style).
hit verb (EFFECT) B2 [ T ]
to have an unpleasant or negative effect on a person or thing:
Production has been badly hit by the strike.
Demand for transatlantic flights has been hit by fears of terrorist
attacks.
Hit (Cambridge Dictionary)
You could use 'hit' in a positive way:
When the news that I has passed the exam hit me, I started jumping for
joy.