This is a great question, because I have been asked this from multiple people - Jews and non-Jews. My answer, of course, is different to each of them, but I'll mention the commonalities.
First of all, before answering, I think you really need to analyze well the intention of the questioner, as you would with any question. Only if the questioner is really willing to hear your opinion as is, even if he disagrees with your answer, in whole or part, should you attempt to answer. Someone who is hostile or is trying to test or trick you should outrightly be ignored, in terms of your answering.
Assuming a positive reception, I think your answer can be quite simple. The essential style of the Talmud is the Socratic learning method. Many people, today, don't learn this way, but it has proven to be a very effective learning method. (I'll see if I can include a link to an article about this, for your interest.) The idea behind the Socratic method is that someone makes a statement, questions are asked and proofs to support or refute that statement are made. Another answer is usually given, and this cycle repeats.
The joy of experiencing this learning method is demonstrated by the fact that almost everyone's opinion is valued. And, a statement is rarely taken and accepted at face value. People can question it, analyze it, argue with it, etc.
In current U.S. society, and much of the world, even though the media has a huge influence on forcing people to think a certain way, in reality, when you speak to the average person, they really treasure their own way of thinking, and like to debate an idea. If you want to go further, you can explain that the Talmud does not really subscribe to "political correctness", and, in a sense, that's probably the greatest pleasure. Not only does the Talmud, itself, encourage debate within its own pages, but, the study of their debate can be debated among the very people who study it.
In terms of answering the question, how can people yell at each other yet still walk away friends - that's exactly the very joy that the Talmud, itself, encourages. What's sad about today's society, esp. in business and offices is that knowledge is "compartmentalized". The way people get ahead is by keeping knowledge to themselves and not sharing it with others. G-d forbid, if someone should know more than you, they will outmaneuver you and you might lose your job. (Whether this is true in a specific job, is not my point, here. Often, this is a fear than reality.) The Talmud, has no such notions of such "knowledge competition". When two friends sit together learning toegther and debating a page of Talmud, they are sharing knowledge. This, in itself, is one of the most visible demonstrations of unselfishness and friendship that I can think of! Each person is willing to share his knowledge with the other with the full intention of making the other more knowledgable.