What follows is a very nebulous question. I just seek for some help in understanding a "technique" which has proven itself very powerful.
I noticed that very often generating series appear in Algebraic Geometry. I am wondering: why can they be a so powerful tool? What is their secret?
What is actually happening when we have computed some numbers, enumerative invariants, classes in a Grothendieck ring... whatever data we are interested in, and we put them all together in a generating series?
Example. Think about Witten's conjecture. One can compute Gromov-Witten invariants $$\int_{\overline M_{g,n}}\psi_1^{a_1}\cup\dots\cup\psi_n^{a_n}$$ and put them all together in the generating series $$F_g=\sum_{n\geq 0}\frac{1}{n!}\sum_{a_1,\dots,a_n} \Big(\int_{\overline M_{g,n}}\psi_1^{a_1}\cup\dots\cup\psi_n^{a_n}\Big)t_{a_1}\dots t_{a_n}.$$ Not yet satisfied, one takes the generating series over all genera $F=\sum_g F_gh^{2g-2}$ and then the exponential $e^F$ of $F$. Witten's conjecture is equivalent to $L_ne^F=0$ for all $n$, where $L_n$ are certain differential operators.
I am sorry for the vagueness of the above. Any insight and concrete example is very welcome.