As an example of application of the Adams spectral sequence I've encountered the computation of the stable homotopy groups of the sphere. This spectral sequence says that $\textrm{Ext}^{s,t}_\mathcal{A}(H\mathbb{F}_p^*\mathbb{S}, H\mathbb{F}_p^*\mathbb{S})$ converges to $\pi_{t-s} \textrm{map}(\mathbb{S}, \mathbb{S})$, where $\textrm{map}(E,F)$ denotes the spectrum $\{\textrm{Hom}_{\textrm{Sp}}(E, \Sigma^n F)\}_{n \geq 0}$ for two spectra $E$ and $F$.
It is claimed that $H \mathbb{F}_p^* \mathbb{S} = \mathbb{F}_p$ and $\pi_{t-s}\textrm{map}(\mathbb{S}, \mathbb{S}) = \pi_{t-s}^S$, the stable homotopy groups of the spheres.
I would appreciate if someone could explain why these two equalities hold. I'm more or less familiarized with the language of the infinity-category of spectra.