What role does Stella play in the conflict of the story? What is the significance of Blanche trying to convince Stella to leave Stanley?
Why is Stanley so bent on discovering and revealing the truth? In what sense is and is not Blanche a liar?
I like the previous thoughts. I would also add that Blanche might be seeing this as the last vestige of hope to try to bring affirmation to her own existence. If we consider this from Blanche's point of view, it makes some sense. Dislodged from her own consciousness and her own conception of understanding what is right and wrong, Blanche's desire to convince Stella of her perception of Stanley might bring some aspect of order back to her world. Without Stella's affirmation, Blanche is cut adrift in a world that makes no sense to her. Stella might just be that one link to something where order and her own sense of right and wrong is present. In terms of Stanley's compulsion to prove to Stella what he sees Blanche as, there can be many possible explanations. He simply might believe that Blanche is a phony and he wants to show that to her sister, his wife. Perhaps, Stanley is threatened by Blanche and sees her as a threat. Recall that he referred to Stella as like Blanche in terms of being placed high up "on those columns" of their Southern mansion. In attempting to undo or sever Stella's ties to her background and being subsumed by his own state of being, Blanche represents a direct threat. Another reason might the result of what Blanche would call "deliberate cruelty." Stanley might just be the type of person who enjoys another's suffering. He recognizes that Blanche is who she is and rather than understand and accept it, he seeks to "out" her in front of others and Stella in demonstrating absolute cruelty.
In A Streetcar Named Desire, the central plot and conflict come down to one battle--the battle for Stella. At its heart, or at least at the heart of the plot, the play is a conflict between Blanche and Stanley for Stella.
That's why the scene during which Blanche tries to convince Stella she should leave Stanley is so important, and why it's so important that Stanley hears what's going on. It's also the reason Stanley is so intent on finding out the "truth" about Blanche. At stake is his wife and his way of life. Stella is that which the conflict is about, to answer your question specifically. Stella is the goal.
Of course, Stanley wins. The "immigrant" Pole, though not technically an immigrant, wins out against the old, traditional South. And what enables Stanley to win? Desire. Life at its heartiest, in a sense. Passion. Love. Lust, if you want to use that word, though I, personally, would not.
Blanche does lie, of course, but she is also a bit deluded, employs defense mechanisms in order to cope with reality, tries to hold on to what she and her family once was. Her behaviors are often not something to be admired. But at the same time, she is much like all of us, just trying to find a way to survive and hang on and stay sane and be well-liked, and to find love.