This is an assemblage of known results, I'll try to put a reference for all of them.

* By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $\mathbb{S}_\mathbb{Q}=H\mathbb{Q}$.

* Rationalization is a smashing localization, so that $\pi_*(\mathbb{S}_\mathbb{Q}\otimes E)\cong \pi_*E \otimes_{\mathbb{Z}}\mathbb{Q}$. This follows immediately from the fact that we can write
$$E_\mathbb{Q}=\mathrm{colim}\left(E\xrightarrow{2}E\xrightarrow{3}E\xrightarrow{4}E\xrightarrow{5}\cdots\right)$$
and that tensor (smash) products and homotopy groups commute with filtered colimits of spectra.

* By Schwede-Shipley Morita theory, the category of modules over $H\mathbb{Q}$ is equivalent to the derived category of $\mathbb{Q}$. (Theorem 5.1.6 in [*Stable model categories are categories of modules*](http://www.math.uni-bonn.de/people/schwede/stablemodelcats.pdf)) Moreover under this equivalence homotopy groups correspond to homology groups (because $H\mathbb{Q}$ is sent to $\mathbb{Q}$ in degree 0).

* In the derived category $D(\mathbb{Q})$ for every object $M$ there is an equivalence $M\cong \bigoplus_{n\in\mathbb{Z}} (H_nM)[n]$. This is an easy exercise, using the fact that in $\mathbb{Q}$-vector spaces every short exact sequence splits.

In particular for every spectrum $E$, the spectrum $E\otimes H\mathbb{Q}$ has homotopy groups $\pi_*(E\otimes H\mathbb{Q})\cong \pi_*E\otimes_{\mathbb{Z}}\mathbb{Q}$ and so there is an equivalence of $H\mathbb{Q}$-modules (in particular of spectra)

$$ E\otimes H\mathbb{Q}\cong \bigoplus_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})$$

The particular formula you are referring to follows from the rational homotopy groups of $\mathrm{bstring}$ and the fact that, for $V,W$ rational vector spaces
$$[V[n],W[m]]=\begin{cases}\mathrm{Hom}_\mathbb{Q}(V,W) & \textrm{ if }n=m\\ 0&\textrm{ otherwise}\end{cases}\,.$$