Apologies for the delay - I missed the notification e-mail about your post for some reason. This isn't really a query about the CPT Bootstrap Carousel wordpress plugin as such, more just about the bootstrap carousel, but I'll see if I can get you started..
.appendTo you're moving all of the captions into your div. These are normally held within divs which are moved and displayed by the carousel, but now they're outside of that environment they are no longer controlled by the carousel and all display on top of each other.
To get them to show and hide as you'd like them to, you need to code up some animations yourself manually. Depending on the effect you're going for, you'll probably need to hide them all (
Hopefully that will be enough to get you on your way. If you need any more help I'd recommend putting your question up at stackoverflow.com or somewhere..