The subjects of an experiment are not the only people who can influence the results of an experiment by knowing who belongs to the test group and who is in the control group. Any other people involved whose judgment or behavior might bias the outcome of the experiment can also be an influence. Normally these would be the experimenters themselves. In the radish experiment, there are at least two ways other people might bias the result.
The people who actually give out the radishes (or carrots) each day might say or do things that would tip off the subjects as to which group they were in or otherwise influence their behavior. A researcher's career might benefit considerably from discovering a new way to improve dental health and she would hate to see the experiment fail because of careless behavior on the part of some subjects, so she might be inclined to tell radish eaters to "take good care of those teeth", while not making similar statements to those in the control group. A laboratory assistant might do the same, knowing that his boss would be very pleased if the experiment turned out to be a success.
The outcome might also be influenced by dentists doing the checkups that count the cavities. There may be some judgment involved here, such as whether two problems on the same tooth are one cavity or two, or whether a tiny flaw should be counted as a cavity. Once again, knowledge of whether the subject is in the test or control group could influence the judgment. In other types of experiment, the judgments might be even more subjective and an even greater problem.
These problems can be solved by "blinding" the people involved. People can give out the radishes or carrots without knowing which is the placebo. Dentists can do the checkup without knowing which group their patients are in. When both the subjects and the experimenters are kept from knowing who is assigned to which treatment, the experiment is called "double blind".