Why does the Ancient Mariner tell his story only to that guest in "The Rime of the Ancient Mariner"?

Coleridge wanted his poem to be spoken by a fictitious narrator in order to avoid giving the impression that he was speaking about an experience of his own. He invented the ancient mariner for this purpose. The fact that this eccentric character seems so authentic adds to the verisimilitude of the long tale. He is exactly the sort of person who could have had that sort of strange experience.

Coleridge also felt obliged to invent another character to whom the ancient mariner was telling his story, although he probably could have claimed that the story had been told to him personally. By having the ancient mariner insisting on telling his story to the wedding guest right in the here and now, especially with the wedding ceremony imminent, a sense of immediacy is created which would have been absent if Coleridge simply claimed that he was repeating a story told to him at some time in the past.

Coleridge added drama to his tale by creating conflict between the ancient mariner and the wedding guest. The wedding guest is late for a wedding, and apparently he is someone who is of importance to the ceremony. He does not want to stop and listen to the weird old character's story and keeps protesting that he is late. But the ancient mariner has "strange power of speech" and holds the younger man against his will.

This is a clever way of adding drama to the poem by creating conflict, which is the essence of drama. Part of the conflict is due to what is called a "ticking clock,"  a time factor that gives a sense of urgency. The wedding guest does not merely want to evade this seedy-looking old vagabond (who may be planning to ask for a handout), but he has a pressing engagement which increases his motivation to get away from him.

The poem about the actual voyage is a story within a story. The cover story ends with the wedding guest being allowed to go on his way, a sadder but a wiser man.

Why does the Ancient Mariner tell his story only to that guest in "The Rime of the Ancient Mariner"?

Actually, the Ancient Mariner is driven to tell his story to almost anyone who will listen. He chose the wedding guest because, for whatever reason of his own, the wedding guest couldn't help but listen:

It is an ancient Mariner,

And he stoppeth one of three.

`By thy long beard and glittering eye,

Now wherefore stopp'st thou me ?

The Bridegroom's doors are opened wide,

And I am next of kin ;

The guests are met, the feast is set :

May'st hear the merry din.'

He holds him with his skinny hand,

`There was a ship,' quoth he.

`Hold off ! unhand me, grey-beard loon !'

Eftsoons his hand dropt he.

He holds him with his glittering eye--

The Wedding-Guest stood still,

And listens like a three years' child :

The Mariner hath his will.

The Wedding-Guest sat on a stone :

He cannot choose but hear ;

And thus spake on that ancient man,

The bright-eyed Mariner.

That's it: at this point the wedding guest is spell-bound and the Mariner begins his tale.

Maybe it should asked, Why, of all people, is a wedding guest, in general, chosen? Well, in some ways the tale that is told by the Mariner has the sense of a wedding about it. The Mariner becomes, after much travail, wedded to a new way of thinking about and loving life and the other creatures of the earth.

At first he felt a separation between himself and the albatross and all the "slimy things" that "did crawl with legs Upon the slimy sea." But at last he learned a deep truth:

He prayeth well, who loveth well
Both man and bird and beast.

He prayeth best, who loveth best
All things both great and small ;
For the dear God who loveth us,
He made and loveth all.

What a fitting little sermon for the Mariner and the wedding guest who both missed the more traditional wedding.

Why does the mariner narrate his story in The Rime of the Ancient Mariner?

In part 7, the mariner explains to the wedding guest why he tells his story. The mariner and the ship began to sail home—helped, in part, by the reanimated bodies of the sailors, now controlled by angels—and when the ship sank, the mariner joined a pilot and a hermit in a rowboat. The mariner sought absolution from the hermit, and though it left him "free" when he confessed his tale to him, the mariner is still sometimes filled with the overwhelming need to tell someone his saga. He explains,

Since then, at an uncertain hour, That agony returns: And till my ghastly tale is told, This heart within me burns.

What is the power and purpose of storytelling in "The Rime of the Ancient Mariner"?

The Ancient Mariner, because of the great sufferings he has undergone, has a power of discernment and speech. He can discern or pick out the person who must hear his story. He says he knows this the moment he sees that person's face. As the mariner states:

I have strange power of speech;

That moment that his face I see,

I know the man that must hear me:

To him my tale I teach.

Beyond being able to choose the right face, the mariner has an extraordinary power as a storyteller. At first, the wedding guest wants nothing more to get into the room where the wedding is held, wishing to enjoy the food and partying that is going on there. Yet the Ancient Mariner has the power to hold him spellbound—and we as readers—by his ability to tell a story well.

At the end, the guest states that he is happier to have heard the mariner's story than gone to the wedding—and he has a renewed sense of faith, expressed in his desire to go to church:

O sweeter than the marriage-feast,

'Tis sweeter far to me,

To walk together to the kirk [church]

With a goodly company!

The mariner is driven to tell his story by a strong sense of purpose. He has learned an important life lesson the hard way, and he wants others to learn it before they have to go through what he has endured. He feels a deep compulsion or burning desire to tell his saga. As he states:

Since then, at an uncertain hour,
That agony returns:
And till my ghastly tale is told,
This heart within me burns.

The story he needs to tell is as follows. His killed a friendly albatross with his crossbow for no other reason that he felt like it and he could. He did it on a whim. In doing so, he showed how little he valued God's creation.

He and the rest of the crew were punished for this act. However, once the mariner learned the lesson God was trying to teach him: he was shown mercy. He grasped that:

He prayeth best, who loveth best
All things both great and small;
For the dear God who loveth us,
He made and loveth all.

This is the mariner's purpose: he wants to spread to people the truth that they should love all of God's creation because God created it all.