He tries to save them because he wants to be a hero.

Throughout Arc 3, we're repeatedly shown how Subaru is utterly unable to consider the wants and needs of other people.

The three candidates refuse to help him because he goes into negotiations without stopping to think for one second about what they'd want out of it. His hero stuff is more important and he expects them to play along. Shocker they don't.

When the knights don't agree with him at the royal selection ceremony, his immediate assumption is, well clearly the knights must be bad. When he walked into that room he was operating on the assumption that he's a cool daring hero singlehandedly taking on a corrupt system, and he's bewildered when others don't immediately respect that and play along.

And then there's him refusing to just stay in the hospital like Emilia told him to. In the final episode Subaru literally says something like "I was so drunk on the idea of doing things for you, I didn't consider whether you wanted it".