In addition to your story and the other person's story, every difficult conversation has an invisible third story. By thinking like a mediator, you can describe the problem between the parties in a way that rings true for both sides simultaneously.
The key is learning to describe the gap – or difference – between your story and the other person's story. The third story captures the difference and allows both sides to buy into the same description of the problem: each feels that their story is acknowledged as a legitimate part of the discussion.
You don't have to know the other person's story to include it in initiating the conversation this way. All you have to do is acknowledge that it's there: that there are probably lots of things you don't understand about their perspective, and that one of the reasons you want to talk is that you want to learn more about their view. If they start the conversation, you can take whatever the other person says and use it as their half of a description from the third story.
Most conversations can be initiated from the third conversation to include both perspectives and invite joint exploration. Keeping communication open and understanding the feelings and perspectives involved sends an important message that even when we disagree, we care about each other.
Source: "Difficult Conversations: How to Discuss What Matters Most" by Douglas Stone, Bruce Patton and Sheila Heen