In The Crucible, Arthur Miller did not describe a real illicit affair between Proctor and Abigail. Why would Arthur Miller add this fictional...

On one hand, I think Miller might have added this fictional affair because it provides Abigail with a really concrete reason, a very strong motivation, to continue making accusations: she wants to eliminate John Proctor's wife and have him all to herself. Abigail's willingness to accuse innocent people like Elizabeth really characterizes her as incredibly manipulative, conniving, and malicious.

On the other hand, Miller might have added this fictional affair because it problematizes the character of John Proctor. He is not some perfect and guiltless man; he is a sinner, and he feels terrible guilt for his affair with Abigail. John initially thinks of himself as a fraud as a result of his affair with Abigail, and his struggle to regain his self-respect and sense of his own goodness is a major component of the play.  His internal conflict is, in many ways, even more important than the external conflict. It is, in the end, John's personal redemption that matters most to him (and to Elizabeth)—so much so that he is willing to trade his life in order to retain it. Without his and Abigail's prior relationship, John would likely be much less relatable because he would not have such personal struggle. John's weakness humanizes him.