Well, to put it shortly, it's just because $L^2$ and $L_z$ are two observables that have no $r$ dependence. As such, it makes sense to write down $Y$ in terms of these two. 

But if we'd like to go a little deeper, it is a bit related to representation theory. The fact that $L^2$ and $L_z$ form a complete description of vector rotations (they represent the Lie algebra $\mathfrak{so}(3)$ of the Lie group $\mathrm{SO}(3)$) and therefore are the most natural way to express the angular dependence of a rotation-invariant space in terms of orthonormalizable functions.