On the surface, he could have been referring merely to active listening. You know, to get the most out of a conversation, you need to set your ideas aside, keep your mouth shut and let the other person talk.
That answer is a fine one. I wonder, too, if we should focus on the word, "reasoning" to capitalize on President Lincoln's deeper point. I think he means "reasoning" in the sense of talking with or persuading another person logically and persuasively.
One might conclude that, in order to be the most successful in reasoning with another individual, we need to understand the ideas that the other person holds true. We need to endeavor to glean the point of view and be open to discussing it.
The way we can best do this is to actively listen and try to exhibit an empathetic spirit, even if (especially if) we don't agree with the point or points the other person is embracing.
How can we expect to persuade another to our point of view if we're not willing to be open to discussion and understanding? How can we be effective with our ideas if we don't understand the background and contexts? It can be extremely uncomfortable, yet very effective.
What do you think?