"My older sister does not want to stay at home."
I was surprised, too, as I didn't know 留 fell into this catagory, but there are a few verbs that act this way (住 and 坐 being the other ones I know of). Someone in another forum said that the place comes after certain verbs that are not really action verbs. It's more that the verb describes the manner of being in that place, almost like the place is the object of the verb. Try to think about the difference between
"I write at home." 我在家写。
and "I stay at home." 我留在家。
Being at home is not part of writing (it's more of a separate piece of information), but being at home seems to be part of "staying" at home. I hope that helps.