I am not quite knowledgeable on modular forms, but as part of the definition of a modular form $f$: $$f\left({\frac{az+b}{cz+d}}\right)=(cz+d)^kf(z)$$ $$SL_{2}(\Bbb Z)=\begin{bmatrix}a & b\\c & d\end{bmatrix},\;\; ad-bc\neq0$$
Why is this an important property of modular forms? Does it have something to do with their symmetry in the complex plane?
It helps to know that modular forms came first, and the definition came later. In the theory of Weierstrass elliptic functions, the invariants $g_2, g_3, \Delta$ are modular forms and it was natural to try to find other examples and generalizations of such objects. Once these were found, it was natural to try to characterize them and to make that the definition of modular forms. It has a lot to do with the symmetries of discrete period lattices in the complex plane.