The lyrics of the ballad really narrate the heartwarming behind the moment, especially with the line bittersweet and strange, finding you can change, learning you were wrong.
When they're about to dance, the Beast is visibly (and audibly) nervous, and then Belle guides his hands to their proper place and makes him feel at ease.
During the "Something There" montage, the Beast is helpless with his spoon, so Belle ignores any technical rules of etiquette to pick up the bowl and sip from its edge, which the Beast then imitates with a grateful smile. The reason he's struggling with the spoon in the first place is that he noticed Belle looking at him awkwardly while he was slobbering down his meal like, well, an animal. The bowl drinking is a meeting in the middle moment that defines the pair's growing relationship beautifully.
That he made the effort to eat with said spoon no doubt helped to speed her emerging feelings along.
The Beast saves Belle from a pack of wolves her first night in the castle. While grand and daring and heroic, it's also motivated somewhat by selfishness: if she dies, he has no chance of becoming human again. The first truly selfless act we see from him is when he takes Belle with him to show her the castle library, which he presents to her as a gift. His childlike glee and excitement as he pulls open the drapes, and her expression of astonishment and wonder (matching the audience's), are truly touching, and the first signs that there really is hope for them.
Beast: You came back? Belle: Of course I came back! I couldn't let them...oh, this is all my fault. If only I'd gotten here sooner... Beast: Maybe it's better...it's better this way. Belle: Don't talk like that! You'll be all right. We're together now, everything's going to be fine, you'll see. Beast: At least..I got to see you...one last time. *closes his eyes and dies* Belle: No! No! Please! Please! Please don't leave me... I love you.
And when the Beast transforms back into a Prince, after a scene that pretty much defines Visual Effects of Awesome... even if you're disappointed with how he looks, the joy in his and Belle's expressions and voices is just so contagious and wonderful. And then they share a kiss that literally sets off fireworks, which transform the castle and its inhabitants back to their original state. Apparently, the screenplay describes it as "The kiss that has waited for so long."
That kiss was so passionate, it made it go from night-time to full-blown daylight.
That Belle hesitates, despite him turning into a handsome prince, until she looks into his eyes. That's what convinces Belle it's him.
It's just as wonderful in the stage shows; often the Prince will feel his head and body, hardly able to believe that the curse is broken, and is so happy he can hardly speak at first.
As the final dance was recycled from Sleeping Beauty, Belle and her prince go in for one more kiss just like Aurora and Phillip, only here it's a blink and you'll miss it moment, as it happens just as the stained glass depiction of them is fading in.
The first "You came back!" is a sweet moment as well.
Though not technically part of the movie, the dedication to lyricist Howard Ashman, who died during production of the movie, at the end of the credits was certainly heartwarming.
To our friend, Howard Who gave a mermaid her voice And a beast his soul, We will be forever grateful.
The simple moment when Belle thanks Beast for saving her life. It's the look of shock on his face as he hears her sincerely thanking him that really drives home this moment is the starting point of his redemption. Him saying "You're welcome" as we pan away from the cozy room is icing on the cake.
Belle's entire relationship with her father, like how she tries to cheer him up, she tries her best to save him and risks her life for him and he for her, and how she leaves the castle to care for him when he's sick and does her best to prove that the Beast is real so he won't get thrown into the asylum.
From the stage show, No Matter What. It really shows how much Belle and Maurice love each other.
You're never strange, don't ever change. You're all I've got, no matter what.
And also from the stage show, "Home (Reprise)."
Cheer up, child. It'll turn out all right in the end, you'll see.
All of the enchanted objects' delight at being changed back, especially Chip, who may have been kind of cute as an object, but is just cute as a button as a boy.
The relationship that all the servants have towards the Beast. Here is this spoiled brat that has gotten them morphed into household appliances, and yet every day they are there for him, trying to help him see the error of his ways and change himself for the better. Never once do they blame the Beast for their predicament, and in fact they actually put their lives on the line to protect him by driving off the attackers in the climactic battle despite the fact that they can be injured and/or killed (as was shown by the feather duster and Lumiere).
Not to mention the Prince's effervescent glee at seeing his friends human again.
Prince: Lumiere! Cogsworth! Mrs. Potts!? Oh, look at us!
Cogsworth pulling a Big Damn Heroes for Lumiere and Lumiere pulling one for Babette. I just thought it was very sweet, especially when Lumiere catches her and they both exchange an adorable expression.
The Beast showing Belle the enormous library in his castle. Her look of amazement seals it.
If a modern person understands that only the upperclass of pre-21st century could understand how expensive even a small library could be it's pretty heartwarming to imagine how elated Belle felt upon seeing Adam's enormous library.
When the Beast has Gaston dangling over the ledge, he almost kills him...then decides to spare his life. It's a small moment, but it shows how much the Beast has changed: if the confrontation had happened early on, he probably wouldn't hesitate to kill Gaston, but after spending time with Belle, he's softened enough to overcome his anger and choose to spare a life, rather then take it.