This question has been essentially answered in the comments. 

(Tilson) We regard a commutative ring as an $E_\infty$ spectrum via the EM functor $H$. This is definitely what Jacob is doing. One could also use associative rings and $A_\infty$ spectra for what follows.

(Wilson) Many of the correspondences between algebra and stable homotopy theory are described in Chapter 7 in Lurie's higher algebra book. 

(Muro) The correspondence between algebras/modules and the associated EM-spectra is laid out in  math.uic.edu/~bshipley/zdga17.pdf (Cor 2.15) which depends on her paper with Stefan Schwede "Equivalences of monoidal model categories."

It is a bit technical, but is easier to work out the correspondence if you restrict to non-negatively graded $\mathbb{Z}$-chain complexes and connective $H\mathbb{Z}$-modules. The correspondence can be spread into two stages:
1) Use the Dold-Kan correspondence to move between chain complexes and simplicial abelian groups.
2) Take the geometric realization of your simplicial abelian group which is a topological abelian group and hence an infinite loop space, so we can take its associated connective spectrum (by repeatedly applying the bar construction). The fact that geometric realization preserves products can then be used to see that this spectrum is an $H\mathbb{Z}$-module.

Now given an $H\mathbb{Z}$-module $M$ we can form the associated simplicial abelian group $\pi_0(H\mathbb{Z}-mod(H\mathbb{Z}\wedge\Sigma^\infty_+ \Delta^i, M))$ to go back. 

This equivalence induces an equivalence their associated stable infinity categories.