What is the relationship between Frances and Robert in Chapter 5-6 of The Sun Also Rises?
Robert feels like he "owes" Frances because she stuck by him when he was not so successful. She was involved in his failed attempt to be a magazine editor and when the magazine failed, it was she who urged him to go to Europe where he could write. After he becomes successful, he finds that other women are throwing themselves at him, and he grows bored with Frances. She is not getting any younger, and she wants him to marry her, but he hesitates at first. In chapter 5, Robert wants Jake to go to South America with him, but Jake refuses, telling Robert to take Frances. Robert tells Jake that he can't just tell her "to go to hell" because he has "certain obligations" towards her. Frances is described as a forceful woman, whereas Robert is insecure.
In chapter 6, Frances has a long conversation -- kind of a monologue actually - with Jake and Robert, in which she tells Jake how Robert is paying her to get rid of her. She is forced to go to England to stay with some friends. She has been with Robert for three years, and he still has not married her, she had wanted to have children, and now he does not want to marry her so she is bitter. Robert does not defend himself other than to interject, off and on, "don't say things like that, Frances" - but he is guilty and he knows that he has not done right by her, that is why he really does not defend himself. He knows that what she is saying is right.
When Robert was down and out, and not feeling too secure about himself, Frances stood by him, but now that he is an up and coming writer, he thinks he can do better, so he tells her he wants to live a little more, experience life.
I highly recommend you examine chapter 6 to get some deeper insights into Robert's character. The sparse dialogue reveals much. I have only scratched the surface to get you started.