I have been trying to understand and learn measure theory and Lebesgue integration on my own.
If one thinks in traditional calculus ideas like 'area under the curve', it looks to me, in total, that Lebesgue integration is simply integrating along the Y axis but using a measure-theoretic idea for the length of each 'slice' of dy.
In other words, traditional Riemann integration involves a limiting process of summing the areas under the curve of a set of rectangles that become dx wide, with a height of f(x). If. somehow, one could take out a countably infinite number of regions of the domain that have measure less than the range of integration, the Riemann approach would fail due to the need for an infinite number of integrals.
But it seems the Lebesgue integral solves this by integrating, if you will, along the Y axis. The width of the 'rectangles' now becomes their measure. Thus, if somehow a countably infinite number of cuts happened on the X axis to make any interval $[a,b]$ have measure $\frac{2}{3}(b-a)$, the Lebesgue integral would reduce the AOC by the appropriate 1/3rd tnrough integrating "vertically" with the measure for the width at each value of y.
Is there more to this than that?
Your interpretation is basically correct, and in fact mentioned by Lebesgue himself in his dissertation. However, there is a bit "more to this than that": the class of functions that can be integrated is not only significantly enlarged ("most" Lebesgue integrable functions are not Riemann integrable, but every Riemann integrable function is Lebesgue integrable), but along with the new theory comes a remarkable suite of convergence theorems, which pretty much forms the foundation of modern functional analysis.