I know it seems strange to think that Kylo Ren could be afraid of Rey, when he is over six feet tall and strong with the Force. While she is inexperienced.
But there are several things in the movie and especially in the novelization that suggests that he is.
This is what he said to her when he met her in the forest from the novelization: "So afraid, yet I am the one who should be scared. You shot first."
Later ... when he was trying to get the map from Rey, he said: "Don't be afraid."
Fear. That's not what she felt. She wasn't afraid of him. She knew what he could do and was trying to stop him. No. He was talking about himself. What he felt. His weaknesses. Fear was the portal into his mind.
... His mind was a turbulent ocean of fear.
Why do I think Kylo Ren would be afraid of Rey?
Why would Kylo Ren be afraid of anyone?
Because they could stop him, that's the only reason he has to fear anyone.
But these quotes came from a part of the story before he knew that she had the Force, before he knew that she had any power to defeat him.
I think he was afraid because he knew that she could destroy him (Kylo Ren) and bring Ben Solo back.
But I also think he was afraid of her because of the compassion he felt for her, and that made him feel weak. And his deepest fear was not being as strong as Darth Vader. He felt that he wasn't because of the compassion he felt. What he didn't know was that it was Anakin's compassion that led him to the Dark Side.