why it's so important to structure our arguments properly

Successful politicians craft their arguments very carefully. They know what to say and what not to say, and the winners are very disciplined about it. Bill Clinton comes to mind.
Politics is largely the art of charlatanism. We don’t want to do that. I won’t do it, which is one of the reasons I never went into politics, which I had wanted to do from an early age. We need to make arguments that are intellectually sound; craft arguments and frame issues in a way that either convinces people to join us, or at the very least leaves them understanding, and feeling, that we know what we are doing, are secure about it, and are advocating things that make sense.
As one example, I have a colleague in the law, a fellow attorney who practices on the other side. I am a plaintiff’s lawyer and he represents defendants. We were preparing the exhibits for a case we were about to try together, as adversaries. (Good attorneys work together collegially on the mundane stuff and save the battles for where they belong.) Richard says he is a Bible-believing Christian. He listens regularly to Christian radio. He is extremely intelligent, a fierce adversary, and in his field of product liability, he adheres to scientific method like a champ. As we were marking exhibits, I pointed out that there is no evidence for the existence of a supreme being (God). As the discussion went forward, I argued that position is more consistent with the stated Christian value of humility than is claiming to know the answer to the ultimate questions of reality. He said (I paraphrase), “there’s nothing humble about claiming to know there’s no God.” Because I had framed the issue properly, and stuck to what I could defend - something I do deliberately and with foresight as to where the various arguments can go - my response was simple and compelling: “I didn’t say there was no God; I said there was no evidence of a God.” Richard looked at me and nodded. I didn’t convert him but he knew I had won the point. Let me be clear. This is an intelligent man. The argument on that point was over, and I had won. Had I not approached it in that way, I could not have won that tiny victory.
We need to win tiny victories and we need to do it consistently. That’s why we have to structure our arguments about religion and secularism like Bill Clinton structures his political arguments. One of the great TV journalists (wish they were still around!) once said that Clinton was the most intelligent president he had ever interviewed. “I always had the feeling that Bill Clinton was six questions ahead of me,” he said. That’s what we need to do.

How I wish I could agree with you, PC. You have a subtle assumption in your argument. That is, that the other person understands, appreciates and accepts clear, logical arguments. I’ve seen too many cases where a few tricky critical thinking fallacies won an argument for an audience, even when they were pointed out to them.
The second problem is that one can only track the logic behind premises back so far. Even if they are clearly true to most, at times the opponent will have a different base premise. For example, your friend could have had the basic premise that the bible is TRUE. While we recognize that there’s no way to prove that, if it’s locked into the other person’s belief system, then it can follow logically that the christian god does exist.
While I liked Clinton and thought he was extremely bright, he did occasionally use clever critical thinking fallacies in his arguments.
Occam

I don’t disagree with that, Occam. We need to understand our audience. However and as an example, we cannot maintain both that there is no god (as an affirmative claim) and that the existence of a god or gods is not falsifiable. Even relatively unsophisticated audiences will catch the contradiction sooner or later. So I’ve never said “there is no God.” I’ve said there is no evidence of a god, the world doesn’t look like anyone consciously made it - things like that. But if we let ourselves get polarized, we’ll walk ourselves into trouble and lose our credibility.

I realize that you said there is no evidence of a god, however, dependent on his belief system your friend could have claimed that the bible is the evidence of a god. Of course there are many arguments against that, but if he accepts it as a basic premise not open to question, then, to him, your statement about evidence is obviously false.
Occam

So what? I’m not going to get through to him anyway. We shouldn’t worry about constructing our arguments to please people who won’t listen to reason. We should assume that our audience is open to being reasoned with; if we cannot start from that premise, then there’s no point in trying to communicate with them at all. Eventually, if I see that someone isn’t listening, I’ll end the discussion. But if all I do is end, or never start, discussions, then I’m guaranteed to make no difference.
In addition, reasoning capabilities are a function, in large part, of what the issue is and which side the respective parties are on. Person x may be completely illogical when you try to discuss the Jesus narrative with her but eminently logical when it’s time to poke holes in your arguments. She may block what she does not want to hear, while being entirely open to a critique of our arguments. And her critique of our arguments may be entirely on the money. I don’t want want to give her that opening. And I don’t want you to do it either.

I agree that we can’t use reason to convert the strong believers, theologically, politically, socially, etc., however, I enjoy discussing, not arguing, with them. Even though I have no hope of changing them, their critique of my reasoning helps me work on my views to make them more precise and accurate in my own mind.
Occam

I agree that we can't use reason to convert the strong believers, theologically, politically, socially, etc., however, I enjoy discussing, not arguing, with them. Even though I have no hope of changing them, their critique of my reasoning helps me work on my views to make them more precise and accurate in my own mind. Occam
It would be nice if simply ignoring all unreasonable people somehow made them all go away. It would be nice if we were all so self aware that we would just catch ourselves being unreasonable. In my case, I'm sometimes surprised by a sudden turn in a conversation when someone becomes not just unreasonable but irrational and almost violent. I eventually had to accept that I might not be good at noticing when people are starting to get annoyed with me. I used to just assume that if people were annoyed with me, it was their problem, as long as I was trying to be reasonable. But, in practice, whether I'm reasonable or not, it doesn't always matter. Sometimes the one person you have to deal with, has to be dealt with on their terms.
I agree that we can't use reason to convert the strong believers, theologically, politically, socially, etc., however, I enjoy discussing, not arguing, with them. Even though I have no hope of changing them, their critique of my reasoning helps me work on my views to make them more precise and accurate in my own mind. Occam
Yes, I've had the same experience. More often than I can count, they force me to ask why their arguments don't make sense. Then I go through the process step-by-step and that makes me a better advocate on that issue.

Dr Neil Postman (chairman of the Department of Culture and Communication at NYU)
has given a good answer as to why the vast majority of people (be they politicians or citizens, religious or secular, educated or illiterate )
cannot reason properly.
Television has a big role.
http://www.youtube.com/watch?v=FRabb6_Gr2Y
While some may find the video long, 2:00-5:00 will provide some motivation to watch the whole program.
For those who want a summary
http://www.youtube.com/watch?v=UhSHDuU9Pgc