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 commutative diagrams
$$\require{AMScd} \begin{CD} \Sigma K(\mathbb{F}_p,i) @>{\Sigma f_i}>> \Sigma K(\mathbb{F}_p,i+d)\\ @V{\sigma}VV @V{\sigma}VV \\ K(\mathbb{F}_p,i+1) @>{f_{i+1}}>> K(\mathbb{F}_p,i+d+1) \end{CD}\,.$$
But this is literally the definition of a map of spectra $H\mathbb{F}_p\to \Sigma^dH\mathbb{F}_p$.