There is no explicit explanation given in the poem - The Rime of the Ancient Mariner as to why the Ancient Mariner stopped the Bridegroom's next of Kin to tell him the story.
If the wedding guest was the bridegroom's next of kin, it may be inferred that He was playing the father of the Bridegroom at this event.
And if that be the case, one may conclude that the Mariner tells the story to this man, that he may give the lessons learned to the Bridegroom.
If it is inferred that the Albatross was a symbol representing the bride, and the Mariner the Bridegroom, then the entire poem is a sort of warning being passed from the ancient Mariner (a man who had journeyed through "life") to the Bridegroom to take care of his bride to be and never to treat her badly.
For if he does, the things which the Mariner and his crew suffered may be considered a topology of the things which the Groom will go through if he doesn't treat his bride (the Albatross) with care.
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