I just want both sides to lift up like the one on the left so that I can make a cool paper airplane folding animation. I tried with a plane and got the same issue. I even tried using two loop cuts in the middle and putting them right next to each other.
You have overlapping faces in your model. You need to delete the ones you don't want.
If you use the dots in face select mode as a guide, it should be hard to find and delete them.
It looks like you may also have duplicated vertices, so I would suggest also using the remove doubles tool on your model to get rid of them.