Why is measure theory needed to understand continuous random variables and probability density functions in particular? Prefacing the question with the fact that I have no knowledge of measure theory. I would prefer a conceptual answer, as there already many mathematical ones.
Also, why don't we need measure theory to understand discrete random variables?
 A: You arguably don't need measure theory to understand continuous random variables at all; those are just the random variables which are absolutely continuous with respect to Lebesgue measure.
For most intents and purposes, the Riemann integral is sufficient in that case.
After all, most commonly used probability densities have very nice regularity properties.
Measure theory is needed, for example, when you need to justify things like the existence of sequences of random variables with prescribed joint distributions, or stochastic processes more generally (e.g., try proving that Brownian motion exists without measure theoretic results like the Kolmogorov extension and continuity theorems).
Another benefit of using measure theory is that it unifies the seemingly similar but distinct continuous and discrete worlds, and allows talking about random variables which are neither.
Elementary treatments of probability often duplicate effort by proving a result in the discrete case and then in the continuous case.
Using measure theory, one can sometimes prove both (and more) at the same time with a proof that might better reveal the important factors at play.
Finally, why isn't measure theory needed in the discrete case?
This is arguably because the dominating measure involved (counting measure) is so easy to work with. For one, null sets don't matter, because the only set with zero counting measure is the empty set.
Secondly, most calculations with discrete random variables amount to regular sums (albeit sometimes infinite).
This makes problems involving discrete random variables tractable even with a very limited mathematical toolkit at your disposal.
