Translation:The dancers want to sew their own costumes.
"Si" (and sia sian, siajn, etc.) is used to refer to the subject. You look at the verb, figure out who is doing the action of the verb, and that's what "si" (etc.) refers to. In this case, the verb is "volas kudri" and that's being done by the "dancistoj" -- so ""siajn" in this sentence refers to the dancers. It's the dancers costumes.
When it's possible to use "si" (etc), it basically necessary to do so. In this case, if you had used "iliajn" as you suggested, the meaning would be different. The listener would think "well, clearly it's not the dancers' costumes, since if it were, the speaker would have said "siajn" - so it must be someone else's costumes. Something along the lines of:
- Estis dancistoj kaj kantistoj, kaj la kantistoj volis kudri iliajn kostumojn."
In this case, "iliajn" would be understood as everybodys, not just those of the singers.
Note also, if the subject of the verb is implied, then "si" would refer to the implied subject. "Si" (etc) cannot be used as part of a subject.
Update I wrote the above two years ago. In the meanwhile, I've also written three blog posts about the topic.