Amor Vincit Omnia
Before the game, Square has stated Rosa was dating Kain. However, by the time the game has started, Cecil and Rosa are in love, and Kain is aware of this. However, Cecil feels unable to act on his feelings in his capacity of Dark Knight. (For some reason, Dark Knights can't date but Dragon Knights can?) This may be an exageration on his part, he may simply be afraid of letting someone love him. He may think he is protecting Rosa by not letting her get to close.
Whatever the case, Cecil believes their love to be doomed from the start.
However, Rosa's not one to put up with that sort of crap. She insists she join the party, her own well-being be damned. She knows her healing skills are invaluble, and she won't let Cecil leave her, unless kidnapped or similar. And she is kidnapped. By Kain. Which puts everything in an awkward place.
Realizing the extent of Kain's jealousy and his feelings for Rosa probably cause Cecil to reflect on his own feelings for Rosa. After nearly losing her, he is certainly more determined to protect her at any cost. He also realizes the depth of his own emotions towards Rosa. This manifests in the scene on the Lunar Whale, where he asks Rosa to leave for her own sake. Rosa refuses, and declares her love for Cecil, who has no choice but to admit his own feelings.
At the end of the game, they are married.
Rosa is often seen as a stereotypical damsel in distress, and while it is true you spend a fair bit of the game rescuing her, every word out of her mouth makes sense and shows that she wasn't meant to be taken lightly. She pretty much bosses Cecil around. She's adamant in her desire to fight and her defense of Kain even after he kidnapped her. She isn't weak.