Here, Isaiah relates a story about a seraph forgiving him of his sins. This is...apparently...relevant? Let's examine it from Nephi's, Moroni's, and Joseph Smith's perspectives.
Nephi knows he's writing this for future generations. Perhaps he realizes that future prophets will read his records and make their own. In that case, it makes a little bit of sense for Nephi--who has questioned his own worthiness--to relate this story of the revered figure of Isaiah struggling with his identity as both a prophet and a fallible man. Maybe Nephi hoped the passage would be a pep talk for future prophets.
Moroni is collecting, abridging, and combining a thousand years of sacred and secular history of his civilization. Why would he include this chapter? Nephi's already given his pick-me-up speech to generations of prophets. Why would Moroni have been inspired to leave this chapter in, considering it was God's plan to have these records be spread across the world and given to millions of people who weren't prophets? All this is available in the book of Isaiah so including it in the Book of Mormon seems like a waste of valuable plate-space (and, later, printing space).
Filler. Quoting the Bible makes his book seem more credible.
On an unrelated note, why exactly do both accounts of Isaiah's story mention that the seraph used tongs to pluck the coal from the fire? It's a supernatural being...is its skin really that sensitive?