If Felicity was apologizing for Oliver’s actions (taking the blame for the breakup) they’d have gotten back together at the end of the episode, because nothing more would be in the way- Oliver wants it, felicity would think ending things was a mistake. Everything’s good. But that’s not what happened. The reason why the breakup seemed so hopeless is because felicity didn’t offer a way out (that was her right, but it locked out any chances of reconciliation) Instead, it was just, ‘you lose Oliver. Things are over’ and that’s also why he didn’t ‘fight for her’ the way people wanted- he had no goal to fight for. The scene in the Argus hospital was felicity changing that. Her understanding was finally realizing what exactly caused the problems between them- why Oliver made his choices. So they talked and finally identified the problem coming between them, and then felicity set her standard. 'Figure out what kind of man you are’. Because then you can trust yourself. Then you can stop sabotaging your relationships. Then we can talk about the possibility of us. That scene was felicity opening up the doors by giving Oliver a destination which he never had before. It’s telling him how to fix what he broke.
Perhaps my favorite scene in 5x20. Felicity assures Oliver that she’s not completely closing the door on them; she just needs time. Quite frankly, she doesn’t have to do that – give him a second chance. She is not obligated to do so. But she provides him one because she knows he’s a good man with a big heart who makes mistakes. Plus, he deserves at least a chance to explain himself.
Unhesitatingly, Oliver assures her that he will not abandon her because he is willing to wait for however long she needs. No matter what. Just look at that relieved expression on her face in the last gif – it’s one of the rarest moments we see her vulnerability because she’s been abandoned by her loved ones before. Even though she hurt him by walking away without giving him a chance to explain, it doesn’t deter his loyalty towards her.
It’s compromise. It’s being fair to one another, respecting one another.