Frank turned his right ear towards the door, the better to hear.
I don't understand the grammar of "the better to hear". Is it elliptical? How should we understand the phrase in this sentence?
I would say the better to [do X] is an idiom that means in order to [do X] better.
It could be a reference to the story,"Little Red Riding Hood". In which the girl makes the statement to the wolf who is disguised as her grandmother,"My, what big ears you have." To which the wolf responds,"The better to hear you with." Or, it could be a reference to some aspect of the character or person Frank having one ear which works better than the other. Or, it could be a poorly structured sentence.