Using empathy works in several ways. First, it defuses a potentially hostile interaction. Anti-violence people are used to being attacked and not understood by advocates of self-defense. Instead of appearing as a violent person, you are now a sympathetic, fellow human being. This may also open the door for a friendly conversation, in which you may each discover that your "opponent" is a person with whom you have some things in common. You may even create an opportunity to dispel some of the misinformation about self-defense and the martial arts that is so prevalent.
This empathy technique is also useful for redirecting, or ending, a heated argument that has become hostile and unproductive. With empathy, you can reframe the argument entirely.
You should not expect any of these approaches to work immediately. With rare exceptions, the anti-violence person is simply not going to "see the light" and thank you for your help. What you are doing is putting tiny chinks into the armor of the person's defenses, or planting seeds that may someday develop into a more open mind or a more rational analysis. This process can take months or years.