When the Any Man says, "Losing all hope was freedom," he is referring to the alienation from the world that he felt in his life, his disillusionment. He, this Any Man, felt his life was so devoid of anything worthwhile that he distanced himself from the world. His alienation from his society lead to his materialism, and his obsessions with decorating his apartment, making it complete. Which kept him from the freedom of living a fulfilling life, being truly alive. Once his apartment is blown up and all of his possessions are lost, and he mourns greatly because his possessions were to him, his life, and his proof that he exists. He begins to understand that he truly doesn't need his belongings through his transformation thanks to Tyler. He doesn't need these things to be free to live his life the way he really wanted to.
He meets this woman, Marla, who has the same general outlook on life; she hates hers too. At first he displays a dislike for her. Then we later realize that she was a positive influence on his progress in his transformation. Marla states her opinion, in one of the first scenes where she is introduced to the