This is the theorem in which Higman proves a lower bound on the number of isomorphism classes of $p$-groups $H$ in which $\Phi(H)$ is central and elementary abelian.
We have a $p$-group $H$ in which $H/\Phi(H)$ is elementary abelian of order $p^r$ and $\Phi(H)$ is central in $H$ and elementary abelian of order $p^R$.
We look at subgroups $N$ of $\Phi(H)$ of order $p^{R-s}$, and we are considering how many different isomorphism classes of quotients $H/N$ we get. To do this he splits the subgroups $N$ into equivalence classes, where each such class contains groups that give isomorphic quotients $H/N$. The displayed formula is $a/b$, where $a$ is the total number of subgroups of $\Phi(H)$ of order $p^{R-s}$ (which is the same as the number of order $p^s$), and $b$ is the order of the automorphism group of $H/\Phi(H)$, which he has proved is an upper bound on the order of each equivalence class.
To prove this upper bound, he says first that, if $H/N_1 \cong H/N_2$, then there is an automorphism of $H$ that maps $N_1$ to $N_2$. So that would give $|{\rm Aut}(H)|$ as an upper bound on the size of the equivalence classes. But, since any automorphism that induces the identity on $H/\Phi(H)$ must also induce the identity on $\Phi(H)$ and hence fix all of the subgroups $N$, we get the smaller upper bound $|{\rm Aut}(H/\Phi(H)|$ for the equivalence class sizes, and that is what he uses as $b$ in the formula.
