You can use the generating function for Schur polynomials
$$\sum_{\lambda}s_\lambda(x_1,\dots,x_m)s_\lambda(y_1,\dots,y_n)=\prod_{i=1}^m\prod_{j=1}^n\frac 1{1-x_iy_j}.$$
Take $x_1=\dots=x_m=x$ and $y_1=\dots=y_n=1$, gives
$$\sum_{\lambda} x^{|\lambda|}s_\lambda(1^m)s_\lambda(1^n)=\frac 1{(1-x)^{mn}}.$$
It seems you take $m=n=N\kappa=N_f$ and $x=h^2$. You have indeed
$$\frac 1{N^2}\log \sum_{\lambda}s_\lambda(1^{N_f})^2 h^{2|\lambda|}=\frac 1{N^2}\log \frac 1{(1-h^2)^{N^2\kappa^2}}=\kappa^2\log \frac 1{1-h^2}.$$

Note that $s_\lambda(1^n)$ are the dimensions of the irreducible representations of $\mathrm{GL}(n,\mathbb C)$, so you can probably also give a representation-theoretic proof.