Anastasia feels like a thorn has been removed from her after learning that Lucien doesn't actually see Eclair the same way she has initially thought.
He expressed his intentions loud and clear - that he only went with the flow of things and decided to put up with the engagement in order to fulfill his dream. At the same time, he wants to use this opportunity to fix his relationship with his father.
However, he also felt bad for taking advantage of such an important thing for a selfish purpose, especially since the other party turned out to be truly in love with him. This kind of arrangement should involve intimacy and affection, after all - things that he can't possibly provide if he doesn't even have those feelings for that person to begin with.
"This is exactly why I wanted to talk with you, Anastasia. I don't know what to do..." Lucien started again as he leaned on the handrails and hung his head low.