Please bear in mind that just telling someone that you know exactly how they feel is an assumption on your part, not reflective feedback. People prefer to find their own solutions to problems instead of having their problems fixed by others. They do, however, want people to help facilitate, not act as problem solvers or solution givers. These techniques will help you in this process. You are best positioned to change someone's mind after you have listened to that person.
People tend to close down and stick to their position until they feel heard.
Trying to convince someone of the strength of your case at a time when they are feeling unheard is like trying to paint a canvas that already has a brightly colored painting on it. Acknowledge others' motivations, feelings, and point of view, even when you don't agree with what they are saying. Your goal is to understand the message, not judge the veracity of what they say. Talk about things they have said that you can agree with. Focus on shared viewpoints as a way of building common ground.