You use the eraser to clean the ceramic rod, I tend to use a green pan scourer pad and detergent on mine as well (it breaks down the oil in the marks). I guess that's why that is in the picture- but I think Andy said he uses his stones dry so the green pad is probably to move the steel filings off the stones.
I have got a big oil stone too, but it takes great skill to free hand sharpen so I Lansky my (good) blades so I don't wreck 'em.