Listen for a long time. Spend time lurking around the viewpoint and resist the urge to comment. So that you can...
1. Get inside the point of view you find repugnant. Find out how that point of view makes sense to the individual holding it. That means setting aside your own presuppositions to adopt theirs temporarily. It means starting from the perspective that their view makes sense to them, is not irrational, is not merely a device to avoid reality or to justify evil or sin.
For instance, if I'm going to look at the pro-choice side of the abortion debate, I have to start by taking pro-choice assumptions: Women deserve the right to choose for themselves whether or not to carry a baby to term as a way to empower women who are at a disadvantage to men because they are the ones who carry babies (men don't), and the baby is not a person until born. These are the starting assumptions. In order to understand the arguments, I have to start there.
Do I have to agree with them? No. I only have to read their point of view with that filter on. Then I can discover how the idea of abortion functions for pro-choice people. I stop arguing and I discover that women have felt held hostage by their reproductive capacities for most centuries. I find out that fathers don't experience consequences for unintended pregnancies carried to term. I learn that the poor are more victimized by unwanted pregnancies than the middle class. I read statistics about women dying on abortion tables when it was illegal... and so on. In short, the presuppositions of the pro-choice side of the debate are the result of a view of history and women's place in it.
I can still not agree fundamentally with abortion (philosophically), but that doesn't prevent me from seeing how this other side, this other point of view coheres, aims to resolve problems my view is less interested in. In fact, I can go so far as to say that the problems the pro-choice view aims to address are important and deserve a deeper look. Simply arguing about whether the unborn are legally persons the way the born are doesn't even address the actual issues pro-choice advocates are concerned about.
Another example: Suicide bombers. To understand why this worldview functions successfully for Middle Eastern Muslim Extremists (note that that is our word for them, not theirs) means to enter and inhabit a world where westernism is seen as the enemy. This shouldn't even be a stretch for most conservative Christians seeing that we have our own culture war in America grounded in religious idealism.
What is a stretch is to understand how and why violence is perceived as a solution. There's no need to simply keep stating that they are crazy. Crazy people can't be changed. They need to be eliminated (hmmmm - sounds a bit like our foreign policy). Yet if we see the world through their eyes, we may come to the realization that what appears irrational from the outside is terrifyingly consistent and a satisfying means of addressing the crisis they perceive.
We might even see some parallels to our own worldview. There are plenty of leaders and pundits who argue that the only solution to the Middle Eastern crisis is the elimination of (fill in the blank) Muslims. We justify our violence as protecting our world, as being "civilized" because it is government endorsed. They justify theirs as defending their religion and culture against bullies.
To understand the worldview just means allowing yourself to get inside the sandals of the other long enough to see that they have successfully justified to themselves what they believe.
So that's step one.
2. Restating the other view using the vocabulary of advocates is a great place to start when attempting to understand a viewpoint you don't hold. This is how I've understood the benefits of reformed theology after listening to tapes, sermons, reading books, interacting online and sharing a mission field with reformed believers:
“The sovereignty of God leads to a place of yieldedness, gratitude and peace insofar as you let go of the burden to make all that is wrong with the world right. Salvation by election as a free gift feels like a mind-blowing, deeply satisfying, cleansing experience that goes beyond working for salvation. Reformed Christians see the glory of God manifested in their midst when people discover the gap that exists between themselves and the holiness of God, which moves them to awe, which leads them to humble worship. They then celebrate that holiness through creeds, through sacraments, through covenantal community, through the recitation of God’s greatness as contrasted with human frailty. This theological system makes sense of a chaotic world, puts God at the center and relieves human beings of the burden of their sinfulness.”Even if I don't experience Calvinism this way, it doesn't prevent me from acknowledging that many, many people in fact do! And as Mariam said in a comment a few weeks ago, there is a zen-like detachment from the suffering of the world (because of trust in God) that leads to peace and acceptance which is deeply appealing to the human condition.
It is also possible to recognize that the idea of grace, for instance, is a revolutionary, universally powerful meme that has had profound consequences for good in the centuries since the reformation, without also believing in all five points of the TULIP. Restating has a way of revealing one's own agenda (if you can't do it without sneaking in sideways attacks, you haven't let go of your agenda yet). It also helps the other person to feel heard.
3. Agreeing to disagree is not satisfying. What feels better is to know that the other person has heard how what you value functions in your life and that you are honoring the truth as best as you currently understand it. The problem occurs when two different points of view must cooperate in a single context (a church, a marriage, a family, a business).
The scale of difference in point of view invariably leads to relational crisis. And it usually means changes of some kind. What is never helpful in any setting is pushing one set of presumptions/assumptions as normative for everyone when that is no longer the case for one or more. The more coercive the environment becomes, the harder it is to keep the relationships together.
We've managed to keep our marriage together when the big changes occurred, but I wasn't able to stay in our church. I've found a place in our homeschooling community, but I had to let my online community go. I've gotten closer to some of my old friends and lost some. It's always tough to face changes in relationship due to changes in beliefs. It's made me want to honor the centrality of others people's belief systems even when I don't agree with them... and I avoid intruding on contexts where my beliefs aren't welcome.
These are a few of the things that have worked for me.