Recall that by representability of cohomology plus the Yoneda lemma, a cohomology operation $H^i→H^j$ is the same thing as a map
$$ K(\mathbb{F}_p,i)→K(\mathbb{F}_p,j)\,.$$
Moreover, the suspension isomorphism $\sigma:H^i(X)\cong H^{i+1}(\Sigma X)$ is implemented by the counit of the suspension-loopspace adjunction
$$\sigma: ΣK(\mathbb{F}_p,i)\cong ΣΩK(\mathbb{F}_p,i+1)→K(\mathbb{F}_p,i+1)$$
by sending a map $X\to K(\mathbb{F}_p,i)$ to $\Sigma X\to \Sigma K(\mathbb{F}_p,i)\to K(\mathbb{F}_p,i+1)$.

Putting all together, a stable cohomology operation $f:H^\ast→H^{\ast+d}$ is a collection of maps
$$\{f_i:K(\mathbb{F}_p,i)→K(\mathbb{F}_p,i+d)\}_{i\ge 0}$$
together with a family of homotopy commutative diagrams

$$\require{AMScd}
\begin{CD}
\Sigma K(\mathbb{F}_p,i) @>{\Sigma f_i}>> \Sigma K(\mathbb{F}_p,i+d)\\
@V{\sigma}VV @V{\sigma}VV \\
K(\mathbb{F}_p,i+1) @>{f_{i+1}}>> K(\mathbb{F}_p,i+d+1)
\end{CD}\,.$$

So, the group of stable cohomology operations of degree $d$ is precisely
$$\mathscr{A}^d=\lim [K(\mathbb{F}_p,i),K(\mathbb{F}_p,i+d)]\cong \lim H^{i+d}(K(\mathbb{F}_p,i)$$

On the other hand, we have (basically by definition)
$$\mathrm{Map}(H\mathbb{F}_p,\Sigma^dH\mathbb{F}_p)\cong \mathrm{ho}\lim \mathrm{Map}(K(\mathbb{F}_p,i),K(\mathbb{F}_p,i+d))\,.$$
where with $\mathrm{Map}$ I'm denoting the space of maps of pointed spaces and of spectra respectively. So we have a Milnor exact sequence
$$0\to \lim{}^1 H^{i+d-1}(K(\mathbb{F}_p,i))\to [H\mathbb{F}_p,\Sigma^d H\mathbb{F}_p] \to \mathscr{A}^d\to 0$$
So the result you want follows if we can show that the $\lim{}^1$-term is trivial. This is easy enough to do directly (we know all cohomology groups involved), although I feel there should be some proof that works without this computational input.

**EDIT:** Dylan Wilson in the comments notes that we can say that the $\lim{}^1$ vanishes simply because all $\mathbb{F}_p$-vector spaces in the direct system are finite dimensional, without need for a detailed computation of the cohomology of EM spaces.