I tried taking the screws out, but they just turned and turned without ever coming out so I just pried the whole thing up. Pop!
The rest of the carpet was just held in place with rusty old staples every six inches or so along the edge and those popped out with almost no resistance. I just grabbed the carpet and pulled.
It only took me about 10 minutes all together. Then I rolled that dirty carpet up and threw it away! Success!
It already smells so much better in there. Not bad for spending about as much time as it takes me to go to the bathroom.
I'm a slow pooper.