When Mama says Walter is "kind of like a rainbow after the rain," she is referring to the troubles he has recently been through. Wanting to show trust in him and affirm his manhood, Mama gives Walter both his own and Beneatha's share of the insurance money, telling him to save Beneatha's portion for her medical school expenses. Instead, Walter allows himself to be cheated out of all of the money. He lets his desires and fantasies about prosperity and owning his own business overrule his reason and prudence. The loss of the money is his low point, the "rain" that comes into his life.
However, when Walter stands up for the idea of his family moving into the house in the white neighborhood, he does, as Mama says, "finally come into his manhood." Walter shows that he is thinking about the family as a whole and not just himself when he refuses the extra money Mr. Linder offers the family for not moving. Walter realizes that the family needs to take a stand for what they believe in, which is their right to live in a white neighborhood, and he needs his son to see him doing this. Walter's father, through his hard work, earned the house for his family, and Walter is not going to let that go. The "rainbow" after the rain is Walter's decision to stand up for their rights, even knowing how difficult their lives might be in the new neighborhood.