Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I know the function's use and value to Ramanujan at the time.

All questions of the form "Why was such a mathematician interested in such a subject?" are difficult, and have a tendency to becomes metaphysical ("why are we doing mathematics in general?", and then "why are we here, anyway?"), but they are even harder when they concern Ramanujan, who had a nonstandard mathematical formation and a very original mind. You're right that Ramanujan could not have been influenced in his interest in the tau sequence by our modern vision of this function as the prototype of the general sequence of coefficients of modular forms, with all the connections to algebraic geometry and number theory that are now familiar, since on the contrary the modern theory of modular forms was developed by Mordell and Hecke after and motivated by Ramanujan's results and questions about the $\tau$ function. So how could Ramanujan have been interested in the $\tau$function? Well, Ramanujan all his life, and well before he came to England and met Hardy, was interested in $q$series, roughly the study of certain formal power series in one variable ($q$), and he valued very much his results that took the form of nontrivial identities between two $q$series. It is an old subject, which begins with Euler (for example his "pentagonal number theorem" for $\prod_n (1q^n)$), and is deeply connected to combinatorics, yet it was at the time of Ramanujan (and to some extent still is) a little bit outside of mainstream research. But from this point of view, the study of the $\tau$function, defined as the coefficients of $q \prod (1q^n)^{24}$ fits well into Ramanujan's lifelong interests. And if you're worried about the exponent $24$, remember that Ramanujan dealt with much more baroque formulas. There is another reason to be interested in the $\tau$function, namely that it is the sequence of Fourier coefficients of the Weierstrass Deltafunction $\Delta(z)$. Now the function $\Delta(z)$ was extremely important (and mainstream) in the mathematics of Ramanujan's time, being central in the theory of elliptic functions (or integrals or curves) and interconnected with the work of many mathematicians of the nineteenth century on complex analysis, Riemann surfaces, and algebraic geometry. Ramanujan was not aware of all these connections before he met Hardy (at least according to the latter, who said Ramanujan had almost no knowledge of complex analysis and the theory of elliptic functions) but after that he become very interested in the subject. 


Well, it certainly is difficult (and probably, as far as I am concerned, a bit immodest...) to try to understand the motivations of the greatest mathematical genius of the last millenium, but, long before him, Euler had studied the expansion of $~\prod\limits_{k=1}^{\infty}(1x^k),~$ Dedekind had defined $$~\eta(\tau)=\exp(i\pi\tau/12)\prod\limits_{k=1}^{\infty}(1\exp(2i\pi k \tau)),~$$ and the formula $$~\Delta(\tau)=(2\pi)^{12}\exp(2 i\pi\tau)\prod\limits_{k=1}^{\infty}(1\exp(2i\pi k\tau))^{24}=(2\pi)^{12}\eta^{24}(\tau)$$ $$= (2\pi)^{12}x \prod\limits_{k=1}^{\infty}(1x^k)^{24}~,$$ with $~x=\exp(2 i\pi\tau)~$ was assuredly known. So, Ramanujan, whose computing abilities were truly amazing, maybe considered the expansion of $~x\prod\limits_{k=1}^{\infty}(1x^k)^{24}~$ as beeing worth the effort, and thus discovered that his ``$\tau$function" was multiplicative and seemed to satisfy marvellous identities... But maybe I'm just tasseographing ("tealeaves reading") the mathematical past. 

