"The siblings often play with their cousins."
Translation:La gefratoj ofte ludas kun siaj gekuzoj.
You need to use "sia" as a possessive pronoun when said pronoun (in this case, their in the object position is made in reference to something the subject of the sentence (the siblings = they > their) owns/possesses/has. With "sia" you make sure that you're still talking about "the siblings' cousins" and not someone else's (a different "they"), which is what ends up being implied if you use "ilia" instead.
I have to agree with @comadrejaweasel. The English does not make it explicit whose cousins they are playing with. It may or may not be the siblings who posses them. I agree that in common usage it's probably the siblings who posses them. That being said, "ilia" and "sia" should both be acceptable.
The original Esperanto phrase is more precise than the english translation given in this case. In the Esperanto sentence "Ili ludas kun siaj kuzoj," they are playing with their own cousins, and not with some other children's cousins. The English, "they play with their cousins" could be understood either way. Perhaps, "the siblings often play with their own cousins," would be a more precise translation, though less usual.