What does this equation mean?

The relationship Waples and Do build their estimator around, in the form most commonly used for the single-sample method: $\hat{N}_e \approx \frac{1}{3\,(\hat{r}^2 - \tfrac{1}{S})}$. where $\hat{r}^2$ is the mean squared correlation of allele frequencies across pairs of loci observed in the sample, and S is the sample size, with the 1/S term correcting for the sampling bias that a finite number of genotyped individuals introduces into $\hat{r}^2$ even when the true population $N_e$ is large. The practical consequence practitioners have to internalize: $\hat{r}^2$ is estimated with noise from a finite sample, and that noise itself creates apparent linkage disequilibrium indistinguishable from drift-generated LD unless it is explicitly subtracted — which is exactly what the 1/S term does, and why the estimator is only trustworthy when the sample size and locus count used to compute it are reported alongside the point estimate, not just the final $\hat{N}_e$ .