Why are the Appalachian mountains now located in the middle of the North American plate?

Because of how the world split and all the land was once together after millions of years the lands started breaking off and moving. We even have land today that's going to be moved later in life or land that's already in the process of moving.

The Appalachians formed when North America and Africa collided. In time, the places separated and so much crust was created that the mountains were no longer at the plate boundary.