it's hearwarming because Fi goes from being a very detached and analytic character, to telling Link she loves him in this piece.
at the end of the game, Fi says that her mind will slowly disappear as she remains sealed within the Master Sword as Demise's soul is slowly destroyed. There's nothing that Link can do because she's the only thing keeping Demise inside the Master Sword.
Before this, she would have been unable to confess her feelings for Link because she was programed to only act towards him as a Weapon and its Master. Only when she is no longer bound to him, and they can no longer be together, she can confess her feelings before she surrenders to oblivion.
