Interpreting Bayes' thereom with density functions

Question

I'm studying about Bayes' theorem and according to the theorem:

$$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)}$$

This I can understand where it comes from etc. But if I use Bayes' theorem on density functions:

$$f_{X|Y}(x\mid y) = \frac{f_{Y|X}(y\mid x)f_X(x)}{f_Y(y)}$$

This makes me raise some questions. What does this mean? Why is it true? Is there a proof for this?

Here is another question of mine relating to this one: Understanding how the rules of probability apply to probability density functions