I asked this question on MSE but got no answer. I was advised to put it here. I am sorry if it is not suitable for MO.

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 referring to enumerative geometry, for instance Gromov-Witten theory or Donaldson-Thomas theory. I am wondering: why can they be a so powerful tool? What is their secret, which makes them so helpful even if they look, at a glance, so intractable?

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.

Geometry and physics, Phil. Trans. R. Soc. A 2010, highly recommended) by Atiyah, Dijkgraaf, and Hitchin, dualities in physics, including the one that underlies the enumeration formula you mention, are often [always?] "captured by a generating function that allows two different expansions." This is true of other mathematical formulas as well. In this way generating series are more than just a formal book keeping device for recurrence relations among coefficients. In any case they fully deserve to be called functions rather than power series. – Vesselin Dimitrov Sep 21 '13 at 0:44