The classic source for an answer to your question is probably Carl Sandberg's "Abraham Lincoln The Prairie Years Volume 2 page 210-216; but in answering that question, certain predicates must be explained 1st. Then, as now, New Yorkers considered the rest of the country "flyover country" populated only by ignoramuses, farmers and hicks with a few second-rate men of substance thrown in to keep the whole mess from flying apart. Of the listeners in Cooper Union, Sandberg reports "Some had heard vaguely that this Lincoln person at once fought a duel and killed a man out in Illinois; at any rate, he came from a region of corn fed farmers, steamboat explosions, camp meeting revivals, political barbecues, boom towns and repudiated state canal bonds. Also, they knew this Lincoln had been the 1st man to grapple and get stiff handling to the dramatic and powerful Stephen a Douglas".
Thus, if Lincoln were to have any chance of becoming president, he would need to convince what we would today call the East Coast establishment that he was more than the equal of any New York Republicans. There were those who thought it was a fool's errand.… He was speaking to the elite of New York society, not "since the days of Clay and Webster had [there] been a larger assembly of the intellect and moral culture of the city of New York. It included people who had heard Jenny Lind and Adelina Patti warble, who had seen French and Spanish dancers, who had spoken with PT Barnum and studied his freaks and monstrosities, read the newspapers edited by Greely, Bennett and, and Raymond who believed the undersea cable for instant communication from New York to be to London would be soon repaired.The pick and flower of the New York culture was there."
However they had never heard anyone like Lincoln speak before. But we should return to that point later. In that era, unless spoken by an abolitionist, the word "abolitionist" was almost universally preceded by taking of the Lord's name in vain-a G-- D--- abolitionist. Abolitionists were crazy people who wish to upset the entire standing order the United States. Even though slavery was not permitted in New York, most New York banks held mortgages on slaves which would become worthless upon abolition.
Accordingly Lincoln's talk was largely limited to establishing that the federal government had the right and power to restrict the spread of slavery to the territories. His beginning point for his speech was a quote from an acknowledged giant, Stephen A Douglas, “our fathers when they framed this government under which we live, understood this question [of slavery] just as well and even better than we do now.” Lincoln demolished this statement with a meticulously researched speech he had put vast effort into back in Springfield " going to the state library nearly every day, searching the Congressional Globe, tracking down details of fact in history, reading clippings he and Herndon had made since 1848 from the Charleston Mercury, Richmond Enquirer, Louisville Journal, going back to numbers of the Southern Literary Messenger. He wore out the patience of Herndon with writing and rewriting some parts of the speech" defining these fathers as "39 framers the original Constitution and the 76 members of the Congress who framed the amendments" then proceeded to present a "crisscross of roll calls, quotations, documents an established history connected with the sacred names of early bygone times, to prove the fathers were with the Republican Party view of slavery and against the Democratic position." At this point I must refer you to the speech where you will see the absolute obliteration of Douglas's argument. If you are unduly softhearted, you might almost feel sorry for Douglas. Lincoln conclusively demonstrated that Douglas's argument was not only wrong but was provably, clearly, conclusively completely unfounded.
However, this much could've been established by a pamphlet written by Lincoln, circulated to New York and reprinted in the news papers. There was much more involved in the impact of the speech upon the cream of New York society.
Lincoln was an extraordinary speaker. As described by a Democrat, a hostile listener, Lincoln's delivery was that of "a dangerous man, I tell you a dangerous man! And he makes you believe what he says in spite of yourself.” Even though Lincoln was not only '"slow getting started"… and "There were Republicans who weren't sure they should laugh at him or feel sorry for him.
As he got into his speech there came a change. He was telling them something. It was what they wanted said… His loose hung, dangling sleeves were by now forgotten, by himself and by his listeners. At moments he seemed to have drifted out of mind that there was an audience before him; he was sort of speaking to himself. In the quiet of some moments, the only competing sound was the steady sizzle of the gas lights burning.… His face lights with an inward fire,… “His voice was soft and sympathetic as that of girl’… Not lifted above a tone of average conversation… A peculiar naïveté in his manner and voice produced a strange effect on his audience… Hushed for a moment to a silence like that of the dead.”… “He’s the greatest man since St. Paul”… “The tones, the gestures, the kindling eye, and member of the provoking look defy the reporter’s skill. The vast assembly frequently rang with cheers and sharp applause. No man ever before made such an impression on his 1st appeal to a New York audience.”"
Lincoln was a magnetic, almost hypnotic speaker.
Thus the short answer to your question is that Lincoln's speech was a meticulously researched rebuttal of Stephen A Douglas's argument delivered with such a presentation as very few men in history have ever been capable of, delivered to, and overwhelming, the cream of New York society with its brilliance.