They are used with regular Braille paper. Apart from being thicker and sturdier than ‘normal’ writing paper, there’s nothing special about that paper.
The Braille writing tool basically forces you to punch Braille dots in the paper at the correct positions/distances (interestingly, as you punch the dots from the backside of the paper, you have to write a mirror image of the text)
The photo in https://commons.m.wikimedia.org/wiki/File:IBM_Port-A-Punch.j... doesn’t show anything like that in the tool.
Also, zooming in, I think I see that this doesn’t use standard punch cards, but ones prepared to make it easier to punch out the holes. You can’t see it in the photo, but I would guess that the bottom of the device has holes through which the chaff falls out, possibly with a collection box, just as that of a hole punch. So, you push a punch, probably slightly smaller than the hole to make, on the center of the hole to make, onto the paper, shearing of the paper along pre-weakened lines.