Why did Biff react the way that he did to the discovery of Willy's affair--calling his father a "fake" and rejecting him completely?
We can come to several conclusions as to why Biff reacts the way that he does to the discovery of Willy's affair. To make the most accurate assumption, however, explore how the lives of Willy and Biff intermingled before the discovery of the affair, and what influence, if any, would this have in Biff's life considering his closeness to Willy.
In Death of a Salesman we learn how the character of salesman Willy Loman is a man with interesting characteristics. For one, his philosophy of life is that being good looking and well-liked are the keys to complete success in life. Aside from this shallow idea, he also proposes that, since his son, Biff, is "built like an Adonis", is popular at school, and great in Football, then Biff's life is for sure going to be a success. All of those ideas fester in Willy's mind so strongly that he makes them his canon of living.
Biff becomes involved in the same scheme, not knowing how dangerous and wrong this is. For instance, since Willy does not believe in losing, he instills in Biff the notion that Biff can cheat to get what he wants. On one occasion, when Biff steals a football, Willy praises his "cleverness".
This very dysfunctional admiration causes Willy to build "a character" out of his son, Biff. He over-praises him, over-stimulates him, and protects Biff's ego by showing him only what is great about Biff. He is slowly creating a very conceited, very overconfident and very clueless young man that Willy goes as far as describing:
Like a young god. Hercules- something like that. And the sun, the sun all around him. Remember how he waved to me? Right up from the field...[…] Loman! Loman! Loman! God Almighty, he’ll be great yet. A star like that, magnificent, can never really fade away!
As a result of all this, not only does Biff create too much complaisance about his own worthiness, but he becomes Willy's puppet and allows himself to be manhandled, basically, by Willy's whims. Slowly, Biff becomes enmeshed with Willy and basically loses his identity: everything he does, wants, and plans is based on Willy's view of Biff.
Hence, when Biff discovers that his father, who has built him up, made him up, and put him on a pedestal, is capable of lying, everything that Biff thinks that he knows about his father comes crashing down. Moreover, the admiration is even more lost when he sees that the lies are also going to affect Biff's own mother.
Therefore, what we see in the incident with "the woman", and Biff cutting ties with his father is a symptom of the shock that Biff experiences when he realizes that his "maker", his "demi-god" of a father, is actually a flawed man. It is not only the anger and the disillusion, but also the question on whether everything Willy thought and said about Biff were also lies.