Girard's theory of mimetic desire argues that all desire is mimetic or imitative. In other words, we desire something only because we first see that someone else has it. Our desire is built on envy. We want what others have, especially if that object seems off limits to us.
We can see one of many examples of this in literature in the opening to Tom Sawyer. Tom is able to excite in other boys the desire to paint a fence by doing it himself as if it is a privilege and then denying them the chance to join in until they are begging and bribing him for the chance to do this chore bestowed on him by Aunt Polly as a punishment.
Mimetic desire is very important to love, according to Girard. A man finds a woman desirable, he says, only when another man already wants her.
We can easily see how this plays outs in Streetcar. At first, the earthy Stanley is not particularly sexually attracted to the frail, unrealistic, romantic Blanche. However, as she becomes desirable to his friend Mitch to the point of Mitch wanting to marry her—which is Blanche's ticket out of a desperate situation—she then becomes desirable to Stanley. Further, as Stanley and Blanche's situation deteriorates and Blanche more insistently asserts her superiority to Stanley, putting herself out of Stanley's reach, the more he wants her. He may do what he can to destroy the relationship between Mitch and Blanche to "protect" his friend, but Girard would argue that Stanley's research into Blanche's past is a symptom of his desire for her that has grown out of envy of Mitch's possession of her. Finally, Stanley does what he wants and rapes Blanche, not worrying about its effect on her.
We could also take Girard's theory further: he argues that mimetic desire or envy threatened to tear apart primitive cultures. They therefore developed scapegoating. First, this involved the human sacrifice of an individual who was ritually killed so that the community could release its tensions and aggressions and achieve a new calm and equilibrium. Later, this morphed into a symbolic sacrifice of an innocent animal, the spotless lamb or goat.
It is easy to see Blanche as the innocent who comes into Stanley's society, disrupts it, and must be sacrificed (destroyed) to reestablish peace and order.