The application of Bayes' Theorem to Naive Bayes Classifiers is laid out pretty well on Wikipedia but I thought I'd put forward my understanding of how it's used.
First off, we have the theorem itself:
$$P(C|F) = \frac{P(C){\cdotp}P(F|C)}{P(F)}$$
Or, "the probability of event C occurring given event F occurring is the probability of C multiplied by the probability of F given C, all divided by the probability of F" (or, in the case of classification, we might view C as the class, and F as the feature).