Why is the complement of a language that is not regular also not regular?
Because regular langauges are closed under complementation. That is, if $L$ is regular, so is $\overline{L}$. (Exercise: prove this.)

So, suppose that $L$ is non-regular.  If its complement $\overline{L}$ were regular, then $\overline{\overline{L}}=L$ would also have to be regular.