What is the intuition behind Bayes' theorem and when should I use it?

Bayes' theorem states:

I understand the formula, but I want to build intuition. Can someone explain:

1. Why Bayes' theorem is so important in statistics and machine learning
2. A concrete real-world example where using Bayes' theorem gives a counterintuitive result
3. The difference between the Bayesian and frequentist interpretations of probability

A classic example is medical testing with rare diseases. How does the prior probability $P(A)$ affect the posterior probability $P(A|B)$?