In Act III of A Rasin in the Sun, why does Mama believe that Walter becomes a man when he denies Mr.Linder's offer?
When Walter rejects Lindner's offer for the last time, he does so in a way that is explicitly connected to a pride in his family (and his family history).
Earlier in the play, Walter had denied Lindner's offer for other reasons. He was shocked at the brazen racism of the offer to buy out the Younger family's new house before they moved in. His shock quickly turned to anger and indignity.
Critically, however, Walter's indignity was not attached to a positive dignity rooted in an understanding of the value of his family, its struggles and its strength. Rather, Walter's ability to reject the first offer was an emotional response that came from anger as much as anything else. He also still had hopes of purchasing a liquor store and so had other (hopeful) things on his mind.
When Walter invites Lindner back for the final scene, Walter is prepared to accept the offer. His hope is gone and a bitter despair has taken its place. He seems to feel that the only way for him to assert himself as a man in the household is to accept the offer and agree to participate in the social perspective that puts his family at the mercy of those who hate them, fear them or otherwise place limits on their potential for reasons of skin color.
Walter's bitterness stems from several sources. He resents living a life of servitude as a chauffeur and he regrets his mistaken trust in his friend who stole his money. Also, importantly, Walter struggles to attain a positive sense of who he is in the world. He struggles to attain a sense of dignity and sees his family as others might - - poor, powerless and stuck that way.
It is Lena Younger, Walter's mother, who presents Walter's point of view to him in starkly simple terms.
“Son, I come from five generations of slaves and sharecroppers—but aint nobody in my family never let nobody pay ’em no money that was a way of telling us we wasn’t fit to walk the earth. We aint never been that poor.”
In this speech, Mama challenges her son to recognize the idea that the family is not completely powerless. They have the power to support one another and to define themselves, regardless of what others may say or think or offer by way of a buy-out.
Thus, when Walter rejects Lindner's offer in the end, he is asserting a positive sense of dignity and rejecting more than just a buy-out. He is rejecting a view of his family and himself that would deny them the power to make their own decisions as to where to live, how to spend their money, and who they might become in the world.
The big speech Walter delivers in rejecting Lindner's offer hinges on the idea of embracing a positive sense of family and of taking possession of his own self-definition (to achieve a dignity that was not give to him but earned through agony).
"What I am telling you is that we called you over here to tell you that we are very proud and that this is - - this is my son, who makes the sixth generation of our family in this country, and that we have all thought about your offer and we have decided to move into our house because my father - - my father - - he earned it."
Walter has become mature enough to own himself, in a manner of speaking, and in doing so has come to accept his family as a part of his identity that has value, history and meaning. These are the things Mama has been trying to convince him of and, after some real struggle, Walter has come to believe them.