As discussed in the article, the idea is to define a set of elementary functions, a continuous linear functional on those functions, and then extend by continuity to a larger class of functions.
In this case, the class of elementary functions can be taken as the set of continuous functions on some compact subset of Rn, and the linear functional is the ordinary Riemann integral. This is continuous, in the sense that if a sequence of non-negative continuous functions converges pointwise to zero, then their integrals converge to zero. Thus we can uniquely define an integral for any function which is the pointwise limit of a sequence of continuous functions by taking the limit of their integrals, and apparently this recovers the Lebesgue integral. Alternatively, starting with the step functions (linear combinations of the characteristic functions of intervals) and taking their integral as the area underneath them also gives back the Lebesgue integral.
You can think of this as being analgous to defining a^r for real numbers r by first defining it for rational r by a^{p/q}=\sqrt[q]{a^p}, and then extending to all r by continuity (ie, taking the limit of the function evaluated on rational sequences approaching r).
