Some reasons can be found here. Borel measurable functions are much nicer to deal with. Every continuous function is Borel measurable, but the inverse of a Lebesgue measurable set may not be Lebesgue measurable. Moreover, Borel measurable functions are very well behaved when it comes to conditioning. If $f:(X,\Sigma)\to\mathbb{R}$ is Borel measurable, then a function $g:X\to\mathbb{R}$ is measurable with respect to $(X,\sigma(f))$ if and only if there exists a Borel measurable function $h:\mathbb{R}\to\mathbb{R}$ such that $g=h\circ f$.
On a more conceptual note, the less measurable sets you have in your codomain, the easier it is for a function to be measurable. And if a random variable should represent a random quantity, then all empirically interesting questions can be formulated in terms of simple intervals and their combinations. For, say, statistical applications there is no empirical difference between Borel sets and a Borel set modified by a null set. The distributions (on the reals) commonly applied can usually be given by a cumulative distribution function and such a function essentially determines the probability of intervals.