I recently thought of the analogy that helped my mom understand a bit more of how it feels to be trans and why it is important for us to hear the correct pronouns. I think it helped so I wanted to share it:
Transition is a journey in itself, be it physical or societal. A journey that is taken through and into a world that is different from the old one, running in parallel with it. When someone uses the correct pronouns and the name they are joining you in that world, making that trip easier, reassuring you that you are not alone and supporting it. After all it is hard work to create a world that you will feel comfortable in and can be yourself to the fullest.
When someone refuses to acknowledge that you are taking that trip and instead are sticking with pronouns that they feel comfortable to use then they are living in the old world that is no longer yours. That tends to make relationships more distant, which are harder to maintain as a result.
So please listen when your trans friend or relative is asking you to call them by what they want to be called. They are asking you to be a precious part of their new world.