Actually, it would be more accurate to say that Buddhists say "Everything I think I am is a misunderstanding."
The key point here is that we all have an idea of who we are: I am a person, I am an American, I am a man, whatever. That idea is usually associated with this body and this lifetime. Most people are not conscious of who or what they think they are.
And that idea of who we are separates us from Life and from others.
And yet we are not really separate. We are all interconnected. We are like a wave on the ocean - we have our unique shape, but our fundamental substance (call it life or reality) connects us to everyone and everything.
When we see ourselves as separate, we stop caring about others and may even want to be angry or hurt others. Can you imagine what it would be like if your right hand felt that way. It might slap you around or try to run away. That wouldn't work very well.
When people have unreal ideas about who we are, we act like that runaway hand. When we cut through that idea of being a separate self, we experience the truth of being part of life and everything. We no longer want to hurt others. We want to grow together, to learn, to help. If the left hand gets stuck in a drawer or a door, the right hand naturally works to free it.
Your health is my health. Your freedom is my freedom. Our ideas that we are separate selves are illusions that keep us from understanding that.
When we drop the illusion of separation, we are naturally loving - taking care of others as we take care of ourselves. And we can learn from others more easily, as well, growing in wisdom.