I am wondering what is the easiest way to attach a piece of carpet to a wooden door. The reason for this is that my cats are scratching the bottom of the door, from the bottom of the door to about 2 feet high. Its taking the paint off the door. So, I thought I could put a piece of carpet on there, so when they scratch, they won't hurt the door

Why not purchase a metal door plate to attach to the bottom of the door.
It would add decorative value and resist the scratching better.
Cats like the feel of wood and carpet to sharpen claws on. Brass will not suit their needs and they will move on to other places to scratch, hopefully a scratching post.

But to specifically answer your question, there is some nice metal moulding available at a hardware store to trim the edges and staples can be used to attach the carpet

Alternatively, you could glue to metal plate with contact cement.

I'd go with the simple kick plate, no carpet.