Belle doesn't have a problem with the notion of sacrifice. When it comes to someone she cares about, she's happy to lay it all on the line.
MAURICE: No, Belle! Listen to me: I'm old, I've lived my life!
It's a family trait: Maurice and Belle are willing to sacrifice anything for each other.
MAURICE: Will no one help me?
Maurice, Belle, and ultimately the Beast are all willing to make sacrifices. Contrast that with the townsfolk, who can't be bothered to get up from their beer stools long enough to give a neighbor a hand.
BEAST: Fine! Then go ahead and STARVE!!! If she doesn't eat with me, then she doesn't eat at all!
Belle is happy to make a sacrifice of another sort: giving up food to make a stand on principle. She is NOT going to be ordered around.
BELLE: By the way, thank you…for saving my life.
The scene in the snow is one of the most important in the whole movie because it shows us how the Beast is willing to sacrifice himself to keep Belle safe. And Belle gives up her chance to get away by taking her savior back to the castle.
BEAST: Then you must go to him.
BELLE: What did you say?
BEAST: I release you. You're no longer my prisoner.
And here it is: the Beast's first true gesture of love. He sacrifices everything—his shot at being human again, his time with her, even the happiness of his servants—because Belle needs to go to her father. Of course, she's making a sacrifice too by leaving the Beast and her friends, but it's a sacrifice made for someone else's sake.
BEAST: You…you came back.
BELLE: Of course, I came back…
If the Beast had been paying more attention, he wouldn't have worried so much about losing Belle. His sacrifice is much more meaningful because he never thought she'd return. He wasn't doing it in order to get something back.
BELLE: If only I'd gotten here sooner!
BEAST: Maybe it's better this way.
The Beast views himself as a burden and looks at his death as a sacrifice freeing Belle from her feelings for him. Kinda dark, buddy.
BEAST: At least I got to see you one last time.
This appreciative gesture validates his sacrifice for Belle—a little reward to remind him why he let her go and just how much it means to his soul to have done that.