The mariner has a supernatural power which enables him to recognize a man who will benefit from his tale and who therefore can be stopped and held in a sort of spell while he listens. The mariner also states that he has acquired a strange power of speech as a result of the experiences which he recounts in his tale.
I pass, like night, from land to land;
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.
The mariner is still expiating his sin for killing the albatross. The supernatural entity, i.e., God, who imposed the penance also gave the mariner the two powers--the power of speech and the power to recognize the individuals who would listen to his long tale and learn from it.
Coleridge had to invent these supernatural powers for his story-teller in order to make it plausible that the wedding guest could be compelled to listen to such a long tale, especially when he is next of kin to the bridegroom and has an important part to play in the wedding ceremony. The wedding guest's strong motivation to get to the wedding creates a conflict with the mariner's determination to tell his story to this one man rather than to any other person. This conflict is intended to provide some drama to the mariner's tale. In the end the mariner has won. He has succeeded in holding the wedding guest in his spell until he has finished his whole tale and added the moral.
He prayeth best, who loveth best
All things both great and small;
For the dear God who loveth us,
He made and loveth all.
The wedding guest turns away from the bridegroom's door. It is too late for him to make an appearance, and the mariner's tale has made such a strong impression on him that he can no longer take an interest in such mundane things as wedding ceremonies and wedding feasts.