After the day she's had when Superman comforts Kara in her own native tongue and she hugs him straight away, you can pretty much hear the "dawwww" worldwide.
Wonder Woman and Supergirl hugging and saying their goodbyes, combined with Wonder Woman basically calling Supergirl a little sister and telling her "You'll always have a home here.".
Superman bringing Kara to his family's farm (before Darkseid bursts in and ruins the moment), basically telling her that she can live there, and maybe consider it home-if she wants to basically...Supergirl, during the course of the movie, felt that everybody kept making decisions for her-and she doesn't get a say, not only that, she never truly felt like she belonged anywhere-the only family she's ever known is dead. And Clark is now the only family she has, but while she latched onto the only living family member she had left-she still felt like the type of world-the type of life...he has and lives-is one she doesn't fit or belong in. But the moment they are in the front yard of Clark's old home in quiet and ordinary Smallville...Kara looks down and sees a little family of a mother hen and her chicks walk on by past them...she smiles, thinking she can truly find a home here-and consider Earth her new home and family-with Clark and his adoptive family...commenting, "I think I'm gonna like it here."
Supergirl remembering her mother's name and telling it to Batman, getting a rare smile out of him. It's even more touching when you think about it and take Batman's past/the death of his parents into consideration. It may have seemed odd to ask her if she even remembers her mother's name, but given how the death of Bruce's parents affected him, the idea that someone can't remember the name of their own mother or father-who died right before their very eyes, would obviously bother someone like Bruce, who won't ever forget that moment or the memory of his parents. Telling Batman, the name of her deceased mother was not only a CMOH for Kara's character (heck, Bruce too) on a surface level, but on a deeper one too. At that point, they had truly come to an understanding of one another.
The development of the relationship between Kara and Batman is heartwarming overall. At the very end, Kara (and the audience) waits for Batman's reaction, and when he smiles in approval, Kara is shown really taking in the moment. Prior to this, she was still a bit hesitant if she was as fit for the role as she thought. If she could win Batman's approval, then there wasn't much to worry about at all.
In retrospect, Batman's final gamble to Darkseid - threatening to destroy Apokolips itself with the Hell Spores unless he lets Kara go - is also one. Despite his innate distrust of her from the start which eventually convinces her that he's a Jerk with a Heart of Jerk, he's ultimately willing to threaten to break his one rule to help his best friend get his cousin back. Though Darkseid praises him for being willing to commit genocide to achieve his goal, Bruce's motive was purely based on friendship; he was willing to risk crossing his personal Moral Event Horizon (one he'd not survive if he did) even when threatened with death, all for the sake of his friends. Messed up as it is, it demonstrates in an odd way how big Bruce's heart truly is.
The final scene, after unveiling Kara to the world, Clark takes her to fly in the sky, with her playfully pulling his cape over his head so she can fly past him and the two having a light-hearted race. It's really sweet to see the two play around like that after everything that's happened. At the end of the day, two of the most powerful beings are still able to sit back and have fun in the sky; no matter what angst she went through, Kara is still able to be a plucky young girl.
