A super short answer (I haven't had enough coffee yet this morning to expand): The space of modular forms of a given weight and level form a finite dimensional vector space. This allows one to easilty derive interesting (e.g. combinatorial) relations between their Fourier coefficients, which often are formed from a generating series or encode otherwise meaningful information.
A super short example of why modular forms are cool: using elliptic modular forms and Heegner points gives the (only) method for computing non-torsion rational points on elliptic curves $E/\mathbb{Q}$.
A super great chapter on modular forms: Chapter 1-Elliptic modular forms and their applications- by Zagier in `The 1-2-3 of Modular forms'. It can be read in an evening and answers more of your questions very nicely.
