As per Qiaochu Yuan's comment we need to only understand the space of based maps between $K(A,n)$ with a chosen base point.
The loop-deloop pair of functors establish an equivalence between the categories of $A^\infty$-groups and connected spaces with a base point: $$\Omega: \mathrm{Top}_{*,\ \pi_0=0} \simeq \mathrm{Grp}: \mathbb{B}$$
More generally, $\Omega^n$ and $\mathbb{B}^n$ give an equivalence between $(n-1)$-connected basepointed spaces and $E_n$-groups. This means that $$\mathrm{Top}_*(K(A, n), K(A, n)) = \mathrm{Grp}_{E_n} (A, A) $$
But $A$ is discrete, so all higher coherence data vanishes and a map of $E_n$-groups is the same as a map of set-theoretic groups, thus the answer.