Consider the function $f$ that takes 1 on the rationals in $[0, 1]$ and 0 elsewhere. What's the area under $f$? You can't answer this question with the Riemann integral since the upper and lower limits of the Riemann sums don't agree. With the power of Lebesgue integration, we can say that the area under $f$ is in fact zero, because the rationals should in a sense have "zero length".
Your question is one of the motivations for the development of measure theory and Lebesgue integration. Probability theory is also based on measures, and the question of how to assign probabilities to events and areas to subsets are one and the same.
To even define an integral in the first place, you need to decide on a way to assign lengths to subsets of the real numbers, since this is what you're integrating against. Start with the real line for example: how does one measure the length of a subset of the line? For intervals $(a, b)$ we see that it's length should be $b-a$. But what should be the length of sets like the rational numbers, the irrationals, or the Cantor set? (Interestingly, not all sets can be assigned such a length, see Vitali sets for example). Measure theory tells us how to answer such questions.
The axioms of a measure lay out the properties that we'd like a suitable area/length/volume function to satisfy, and in particular, Lebesgue measure is the measure function you want when thinking of area in the normal way that assigns each interval its length. Measures can actually be thought of as integrals in many cases, which is essentially the content of the Radon-Nikodym theorem.