If you consider uniform probability on an uncountable space, e.g. the unit interval $[0,1]$ then obviously the probability should be invariant under translations (i.e. the probability of being in the subinterval $[a,b]$ is the same as being in the subinterval $[a+r,b+r]$).
It turns out that under the common axioms of mathematics (which include the axiom of choice) there are sets to which we cannot assign probabilities and require that the probability is countably additive.
This is why we work with $\sigma$-algebras, and the Borel one is very suitable because it is generated by the subintervals.