Koans possess a narrative structure insofar as they consist of a brief story, in the form of an dialogue, that is, a conversational exchange of some sort, and the content of the dialogue. Beyond this, the characters in the koan are important. One character we can say is the Buddha, or someone who is assumed to possess Buddha wisdom and another person or persons, who are seeking to realize their Buddha-nature which they've been unable to realize.
The dialogue consists of various allusions which the reader of the koan is trying to pick up since the allusion is a decisive clue to his own true nature. This allusion or implicit reference to his nature might be holding up one finger or the Zen master making a circle with his staff, voicing a phrase or a single word. There are other references, too, which always seem to dumbfound the reader; which appear to go against conventional Buddhist and commonsense understanding.
While the Zen master may seem to be unpredictable, in truth, he is always pointing to the Buddha-nature; but he does this in the conversation by denying our concepts of Buddha-nature, as if forcing us to leap beyond our concepts to where the actual mysterious nature abides. In this respect the koan operates as an antidote to the toxicity of always trying, by way of our intellect, to conceptualize our true nature. Steven Heine sums up the use of the koan nicely.
"According to this standpoint, the words and phrases used in koans are devoid of significance and meaning. They function not as names for ideas, people, or things, but as “no-names” in a sense recalling Lao Tzu's famous saying, “The Tao that is spoken is not the true Tao.” Words are a kind of “poison to counteract poison, ” or a way to fight the fire of inherently misleading discourse through the mechanism of discourse itself. This is a paradoxical, self-deconstructing approach to discourse. Words are used in a penetrating and often devastating fashion as a means to overcome any trace of reliance on language” (Steven Heine, Opening a Mountain: Koans of the Zen Masters, p. 5).
The word in language is always rich with concepts which lead to still more. But it is not the route we must take to behold our true nature. We simply cannot conceptualize our way to the non-conceptual Buddha-nature. We may read about how to swim but entering the water is quite a different matter. Beholding our true nature first requires of us that we accept the fact that our present state of mind, which is like a radio that has never been turned off, is going to be turned off while our eyes are wide open. It is an extraordinary event like no other. We will directly experience pure Mind, which is our Buddha-nature, and really get the meaning of what 'pure' means. It won't be conceptual—it will be all together inexpressible, conceptually.
In a real exchange between a Zen master who has realized his Buddha-nature and a student who has not, the Zen master is always in the antidote mode. He senses the student conceptualizing, out of habit, trying to conceive their true nature, hence, never entering the gateless gate which is an allusion to the One Mind.