It is well proven that alcohol impacts the judgment and executive functions of the brain. When drinking, the emotional centers of the brain begin to control that person's actions. Therefore, it would be that person's emotions (perhaps a guilty conscience) that may drive him to read his Bible when impaired by alcohol. He might feel badly about past actions and seek some relief. On the other hand, if a person is an angry drunk, then that anger may be driving him to find verses about God punishing his enemies. We really don't know the specific reasons a person would read God's Holy Bible when inebriated.
However, God instructs us to approach him in humility and in truth. "God is a spirit: and they that worship him must worship him in spirit and in truth." John 4:23. Perhaps a drunkard may read the words of scriptures but God will only give true understanding when his spirit is pure and truthful.