First I should say that Clausen-Scholze are *not* considering functors which preserve direct sums, but rather all functors. (This is likely why, in the end, one needs to know something about the homology of Eilenberg-MacLane *spaces* rather than the homology of the Eilenberg-MacLane spectrum. That guess/observation I learned from Piotr Pstragowski.)

But if you still want to know about functors that preserve direct sums, then it is indeed true that there's an equivalence $\mathsf{Fun}^{\oplus}(\mathsf{Latt}, \mathsf{D}(\mathbb{Z})) \simeq \mathsf{Mod}_{\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}}$. The 'intuitive' explanation is the following: on the one hand, if we have a functor $F$ on the left hand side, then its value on $\mathbb{Z}$ is some $\mathbb{Z}$-module $M$ and it's equipped with a bunch of extra structure, as a $\mathbb{Z}$-module, coming from the functoriality. This extra structure turns out to be a *second* $\mathbb{Z}$-module structure which `commutes' with the first, hence $M$ gets the structure of a bimodule, or a $\mathbb{Z}\otimes_{\mathbb{S}} \mathbb{Z}$-module. In the other direction, given a bimodule $M$, we can define a functor by $\Lambda \mapsto M \otimes_{\mathbb{Z}} \Lambda$, using the leftover $\mathbb{Z}$-module structure on $M$ to make this a functor from $\mathbb{Z}$-modules to $\mathbb{Z}$-modules.

To be more precise, the equivalence proceeds in three steps.

- There is an inclusion $\mathsf{Latt} \to \mathsf{Mod}_{\mathbb{Z}}^{\mathrm{cn}}$ into connective $\mathbb{Z}$-modules (i.e. chain complexes concentrated in nonnegative degrees). Left Kan extending along this gives an equivalence $\mathsf{Fun}^{\oplus}(\mathsf{Latt}, \mathsf{Mod}_{\mathbb{Z}})\stackrel{\simeq}{\to} \mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}^{\mathrm{cn}}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})$
- There is a canonical equivalence $\mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}^{\mathrm{cn}}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})
\simeq \mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})$.
- Every $\mathbb{Z}$-bimodule, i.e. $\mathbb{Z} \otimes_{\mathbb{S}} \mathbb{Z}$-module, $M$, defines a colimit preserving functor $N \mapsto M\otimes_{\mathbb{Z}} N$. This procedure gives a functor $\mathsf{Mod}_{\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}} \to \mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})$ which turns out to be an equivalence.

Here (1) is a formal consequence fo the fact that Latt gives compact projective generators of $\mathsf{Mod}^{\mathrm{cn}}_{\mathbb{Z}}$ (see, e.g., Higher Topos Theory 5.5.8.22 for the equivalence once you know these are compact projective generators), (2) is a formal consequence fo the fact that $\mathsf{Mod}_{\mathbb{Z}}$ is stable (so there is a unique way to extend a colimit-preserving functor on connective modules to all modules), and (3) can be proven using the Barr-Beck theorem, but a reference (for a stronger statement) would be Higher Algebra 4.8.5.16 (though you'll have to do some translation of the notation...).