Estrangement is the painful experience of being separated from someone or something that you once loved. Whatever happened to cause the separation, estrangement from those you loved and cared for can be a painful experience.
Anne Morrow Lindbergh's quotation is about your estranged self.
When one is a stranger to oneself then one is estranged from others too. If one is out of touch with oneself, then one cannot touch others.
When you are estranged from some part of your own self, the pain can be even greater. Your self is happiest when it is whole. When you are estranged within yourself, you have lost touch with some part of you that you valued. When you are odds with yourself, you feel estranged from who you were.
Self-estrangement can happen when you realize that your hopes and aspirations got waylaid. The part of you that was eager to do something got sidetracked. You lost touch with yourself. And in the process, you became what you do not love.
Self-estrangement can also result from selling out. Selling out means you compromise your integrity to get something you want.