The message is, of course "You can't be too careful about radiation. Where is your radiation protection?".

I assume this is a quote from the film. The badge is the protection. It protects by measuring the total amount of radiation recieved by the wearer. There's a piece of photographic film in there, and how cloudy it gets depends upon the exposure to radiation. It's not some magic device that stops exposure, just a check of accumulated dose. If a wearer is found to have recieved a particularly high dose, they can be checked out medically or their post changed. There's only a mystery here if you don't understand how these things are managed.