Ok, this passage is just incredibly awesome when you really delve into it and dwell on it. At the end of this conversation, Jesus reveals himself to be the Messiah (called Christ). Ok so just imagine how crazy this situation is. If he is truly Christ, then he is the Son of God — as in that he is God, the creator of the universe. Someone with this immense power came to this well and asked a Samaritan woman for a drink of water! He didn't need to, he could have gotten that water himself, but he chose to associate with this Samaritan woman. He didn't introduce himself as Christ, he just came to her as a human man in need. Not only that, but a Jewish man. The kind who would never associate with a Samaritan. This immediately brought curiosity. Looking through the passage, it looks like Jesus never answers any of her questions, but takes a weird detour. In reality, for her to understand why he would ask a Samaritan woman for a drink, Jesus needed her to realise that he is Christ and that he has a deep love, not just for Jews, but for all people!
I feel like she is almost mocking Jesus in the beginning. She must think he's a little bit crazy
Who is this guy? Then he told her things about her that he couldn't know, revealing that he really is something else. This reminds of her of the Messiah's coming and so this is where Jesus decides to reveal himself as Christ. He takes her on a journey of understanding, but it was on her terms. She came to believing in her own personal way.
Why did this Jew ask a Samaritan woman for a drink? Because of His (God's) immense love for her and all people!