I believe these are what you're talking about -
They dampen the noise/shock when you let your foot off the pedal. I can't see the staples in this picture but I'd say you're correct about that.
I just went out and stuck my camera under my '53 and took these pictures.
You can't really tell what holds them on. Looks like there are small holes where some type of staple or stitch goes through. Sorry, I'm not going to pull my nice fitting floor mat back to look from the inside - too hard to get it and my firewall pad back down even again.