Symbolism, Imagery, Allegory
The Main Dancing Princess
Azalea is mad for dancing. It colors how she thinks about herself and helps her understand the world, as we see in this thought she has soon after her mother's death:
But Azalea did know one thing: she was a fast learner. When she fumbled through a dance step it was only a moment before she caught the rhythm and glided back into the motions. If Mother could smooth things over, then she could too. (4.33)
She's thinking about how to smooth things over with her sisters here—which has pretty much nothing to do with dancing—and yet dance is how she makes sense of the problem and finds the confidence to trust herself to be able to fix it.
Being able to dance well symbolizes everything that Azalea likes about herself. She's so excited for her first Yuletide ball that she tells Fairweller (her partner for the opening dance): "I hope you're good at dancing […] Or this ball will be completely ruined" (2.9). Yup—Azalea cares a whole heck of a lot about dancing and places a lot of pride in her skills. No pressure or anything, though, Fairweller.
Azalea isn't just vain though, and her sisters are truly in awe of how good she is on the dance floor. Bramble tells Keeper before he dances the Entwine with Azalea, "Good luck […] Azalea's never been caught. You can try, though" (23.140). And while Azalea's literally never been caught while dancing the Entwine before, we can also see that she's pretty much unmatched in her ability to bust a move. To Azalea, dancing is a skill she's proud of, as well as a big part of her identity.
The Twelve Dancing Princesses
The rest of her sisters are learning to dance, though they're not quite as good as Azalea yet. Dancing is a big deal in their lives right after their mom's death because they're forbidden to do it out of respect for the dead, but dancing also reminds them of their mom and makes them feel closer to her.
As Clover tells the King one time, the girls feel like they can't stop dancing, because it reminds them of their mother. She says:
"But it does help […] Mother would—would dance at night, too. In the ballroom—and—and you were there, and you danced the Entwine, and—you caught her, and she kissed you. On the nose […] I think it was the sweetest thing I've ever seen." (12.65)
The way Clover's talking about dancing here, it symbolizes both the love her mother and father shared, and the emotional bond between the girls and their mother. Dancing is a symbol for coming together in this context (whereas sometimes with Azalea, it's a symbol of how she stands out).
So given the option, of course the girls want to dance. It reminds them of their mother, and of happier times in their family, and it lets them escape the dreariness of mourning temporarily. Dancing thus symbolizes joy, escape, and rebellion. In the immortal words of Lady Gaga, just dance.