It's called a bra, for those of you who have never seen one, let alone heard of what one is. They are used to hold up a woman's breasts. Most women wear them, some choose not to. They come in all sorts of shapes, colors, and sizes. Girls will sometimes wear padded bras to make their breasts look twice their actual size. Others will just stuff their regular bras with socks to do this job.
A wild guess on this ... Dusty is doing some hunting for his owners, and is bringing back the sorts of prey he thinks they like. That would make him both very considerate and very intelligent.
One time a cat hunted me a loaf of bread: he dragged it out of the kitchen by its "neck" (the twisted part of the bag) and set it on my pillow. He's also left me mouse heads, which I have less use for.