This answer assumes some knowledge of bosonic string theory.

In section 7.3 of volume 1 of Polchinski's textbook on string theory, it is proved that the genus 1 amplitude for the bosonic string without any vertex operator insertion is given by

$F_1 \sim \frac{1}{2} \int_F \frac{d\tau d\overline{\tau}}{Im(\tau)} <b(0) \overline{b}(0)\bar{c}(0)c(0)>$

where $b$, $c$ (resp $\overline{b}$, $\overline{c}$) are the left (resp. right) antighost and ghost fields and where the integration is over the fundamental domain $F$ for the action of $SL_2(\mathbb{Z})$ on the upper half-plane, i.e. the moduli space of elliptic curves. Essentially, there is one insertion of the antighosts $b$, $\overline{b}$ because a genus 1 complex curve has one complex moduli and there is one insertion of the ghosts $c$, $\overline{c}$ because a genus 1 complex curve has a one complex dimensional group of automorphisms.

Now the key point is that the algebraic structure of the topological twist of a $N=2$ two dimensional conformal field theory is similar to the algebraic structure appearing in the covariant BRST quantization of the bosonic string. In the topological twisted theory, we are looking at the cohomology of a nilpotent operator $Q=\oint G^+$, the stress tensor is $Q$-exact: $T(z) = \{Q,G^-(z)\}$ and there is a $U(1)$ R-symmetry current $J(z)$ with respect to which $G^+$ and $G^-$ are respectively of charge $1$ and $-1$. In the BRST formulation of the bosonic string, we are looking at the cohomology of the nilpotent BRST operator $Q_B=\oint j_B$ where $j_B$ is the ghost current, the stress tensor is $Q_B$-exact: $T(z) =\{Q_B, b(z)\}$ and there is a $U(1)$ ghost number current $J_B(z) \sim b(z)c(z)$ with respect to which $j_B$ and $b$ have respectively of charge $1$ and $-1$. This suggests the identification of $Q$ with $Q_B$, $G^+$ with $j_B$, $G^-$ with $b$ and $J$ with $J_B$. As we have for the bosonic string

$F_1 \sim \frac{1}{2} \int_F \frac{d\tau d\overline{\tau}}{Im \tau}<J_B(0)\overline{J_B}(0)>$

we have for the topological string that

$F_1 \sim \frac{1}{2} \int_F \frac{d\tau d\overline{\tau}}{Im \tau}<J(0)\overline{J}(0)>$

As $F_L$ and $F_R$ are the charges associated to the currents $J $ and $\overline{J}$, the correlation function $<J(0)\overline{J}(0)>$ is given in operator formalism by $Tr (-1)^F F_L F_R q^{H_L}\overline{q}^{H_R}$.

The preceding derivation works for any topological twist of a $N=(2,2)$ two dimensional conformal field theory. It applies as well to a Calabi-Yau sigma model as to a LG model.