In drug trials, a control group is given a placebo while another group is given the drug being studied. That way, researchers can compare the drug's effectiveness against the placebo's effectiveness.
In a double-blind, placebo-controlled clinical trial, neither the patients nor the researchers know who is getting a placebo and who is getting the treatment. Because patients don't know what they're getting, their belief about what will happen doesn't taint the results. Because the researchers don't know either, they can't hint to patients about what they're getting, and they also won't taint results through their own biased expectations about what the results will be.