Questions about equivalence of Homotopy categories $D(R)$ and $\operatorname{Mod}_{HR}$

Question

Let $R$ a ring and $D(R)$ it's derived category. Two questions:

1. What are homotopy groups of $D(R)$? This terminology is used in this answer.

Conjecture: by definition $D(R)$ is obtained from the space of all chains of $R$-modules supported in positive degrees $Ch_{\ge 0}(R)$ by inverting certain class of morphisms. Therefore the objects of $D(R)$ and $Ch_{\ge 0}(R)$ are the same. Are here maybe simply meant the homotopy groups of the chains?

2. In the linked answer is remarked that the homotopy category $D(R)$ is equivalent to homotopy category $\operatorname{Mod}_{HR}$ of modules over $HR$ (here $HR$ is the Eilenberg-MacLane spectrum of $R$) (considered as subcategory ofspectra).
   Why this mentioned equivalence between these two homotopy categories implies that the homotopy groups of $D(R)$ correspond via this equivalence to homology groups (= $[\mathcal{S}^n, M]$ for sphere spectrum $\mathcal{S}^n$) of $\operatorname{Mod}_{HR}$?