It is not difficult to see that there are only continuum many Borel sets, that is, $2^{\aleph_0}$ many. But there are $2^{2^{\aleph_0}}$ many sets of reals. So most sets of reals are not Borel.
To see that there are only continuum many Borel sets, one can think about how the Borel sets are constructed. We begin with the basic open sets, of which there are countably many, and then systematically close under countable unions, intersections and complements. It follows that the Borel sets are constructed in a hiearchy of length $\omega_1$, and that every Borel set has a construction template, known as a Borel code, that details exactly how it was constructed from the basic open sets. One can think of the Borel code as a well-founded countable tree, whose leaves are labeled with basic open sets and whose other nodes are labeled with union, intersection and complement, meaning that this is the operation to be applied to the children node in order to know which set is coded at the parent node. Every such tree is a countable object, coded by a real. Thus, we can map the reals onto the set of Borel sets, and conversely, so they have the same size, without using AC.
One can also give very concrete examples of sets that are not Borel. For example, the set of reals that code a binary relation on the natural numbers that is a well order is a complete $\Pi^1_1$ set, and cannot be Borel.