Sincere question: Why would you go into a bathroom with rain boots on?
From my extensive manga- and anime-research I can tell you that this is fairly standard onsen-public-bath-cleaning gear. Which I think makes sense. I wear them to clean our pond, too, which clearly is basically the same thing.
I was thinking maybe because if it's your job to regularly clean public bathing areas, no matter how clean you keep them, if you went barefoot, you will end up with athletes foot or verucas etc. And if you wore your regular shoes, getting wet all the time would ruin them - and if they soaked through, might not protect your feet fully either.
