Degree-3 curves on the Calabi–Yau quintic Robbert Dijkgraaf said,1
concerning the simplest
Calabi–Yau space, the quintic:

"A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. The number of degree-two curves was only computed around 1980 and turns out to be much larger: 609,250. But the number of curves of degree three required the help of string theorists."

I gather from OEIS sequence A076912 that
the number is 317,206,375.

Q. Can anyone point me to (or describe) how the degree-3 count
  was settled with "the help of string theorists"?


1Robbert Dijkgraaf. "Quantum Questions Inspire New Math."
Quanta.
The Best Writing in Mathematics 2018, Ed. M. Pitici. p.80. Princeton.
Publisher's link.
 A: The physicists Candelas, De La Ossa, Green, Parkes predicted the virtual  number $n_d$ of rational curves of degree $d$ on a quintic threefold for any $d\geqslant 1$. The numbers $n_d$ are defined by the 'multiple cover' formula
$$\sum_{d=1}^\infty N_d q^d=\sum_{d=1}^\infty \sum_{k=1}^\infty \frac{n_d}{k^3}q^{kd}$$
in terms of other numbers $N_d$, the genus $0$ Gromov-Witten invariants (which, roughly speaking, count maps $f$ – not just embedded curves – from $\mathbf{P}^1$ to the quintic threefold with $f_\ast[\mathbf{P}^1]=d$). The $N_d$ need not be integers, and are in general only rational numbers – e.g. one has $N_2=4876875/8$. How the physicists used a change of variables, the 'mirror map', to equate a generating series of the $n_d$ with an explicitly calculable series associated to the 'mirror' of the quintic is best described elsewhere (e.g. in this book, in various levels of details, or the Perutz article linked in the comments). 
In any case, I would like to point out that it is a subtle question if the numbers $n_d$ are enumerative, that is, $n_d$ is the actual number of rational curves of degree $d$ in the quintic*. For example, the $n_d$ are deformation-invariant (because the $N_d$ are) and so independent of the quintic $X$, but the actual number is certainly not deformation-invariant. Think of the classical case $d=1$ (lines), $n_1=2875$ computed by Schubert. If you take the equation of the quintic $X$ to be $x_0^5+\cdots+x_3^5=0$, then the number of lines on $X$ is not even finite, in fact the moduli space of lines on $X$ has dimension $1$. However, if $X$ is 'generic', then a fairly classical and elementary argument shows that the moduli space of lines is $0$-dimensional and reduced, and the number of points can easily be computed by using the intersection theory of the Grassmannian $\mathrm{Gr}(2,5)$.
For $d>1$ the analysis of the moduli space of rational curves of degree $d$ on $X$ and the intersection theory calculations become much more complicated. For $d=2$ this was carried out by Katz, while the $d=3$ case was handled by Ellingsrud and Strømme. Curiously, the number first computed by Ellingsrud and Strømme in $1991$ was $2682549425$, in disagreement with the number $n_3=317206375$ computed by Candelas, De La Ossa, Green, Parkes. Ellingsrud and Strømme then found an error in their computer program – in a sense, string theory served as a debugger. (The history of the $d=3$ case is described in great detail in this article by P. Galison.)
*Indeed, the number $n_{10}$ is not enumerative.
