The computation is based of the Honda-Tate theorem which states that isogeny classes of abelian varieties over finite fields are completely determined by the characteristic polynomial of their Frobenius automorphism acting on the first $\ell$-adic cohomology group. For a given dimension $g$ and base field of size $q$, a complete list of all Weil polynomials that do occur can be enumerated using a technique developed by Kedlaya [MR:2459990, arXiv:0608104]. In 2016, Dupuy, Kedlaya, Roe and Vincent improved upon Kedlaya's original code to generate these tables and the data they contain.
One may also with to see the article of Kedlaya and Sutherland [MR:3540942, arXiv:1511.06945], where these techniques are used to compute Weil polynomials for K3 surfaces.