A classical example of fuzzy logic in an AI is the expert system Mycin.
Fuzzy logic can be used to deal with probabilities and uncertainties.
If one looks at, for example, predicate logic, then every statement is either true or false. In reality, we don't have this mathematical certainty.
For example, let's say a physician (or expert system) sees a symptom that can be attributed to a few different diseases (say A, B and C). The physician will now attribute a higher likelihood to the possibility of the patient having any of these three diseases. There is no definite true or false statement, but there is a change of weights. This can be reflected in fuzzy logic, but not so easily in symbolic logic.