One version of the question you might be asking is where such equivalences come from in general, as opposed to a proof of this one in particular. The general setup is "derived Morita theory": here is an underived version of the basic result.

As setup, let $C$ be a cocomplete abelian category and let $S$ be an essentially small full subcategory of $C$. This gives a restricted Yoneda embedding

$$Y : C \ni c \mapsto (s \mapsto \text{Hom}(s, c)) \in [S^{op}, \text{Ab}]$$

and we would like to know when this is an equivalence of categories. Roughly speaking this means that $S$ "freely generates" $C$.

**Theorem (Freyd):** The restricted Yoneda embedding is an equivalence of categories if and only if the following conditions hold:

- each object $s \in S$ is compact projective, and
- if $Y(c) = 0$, then $c = 0$.

For a long meandering proof see this blog post. These conditions are unfortunately quite restrictive: for example, for modules compactness is equivalent to being finitely presented, so the only compact projectives are the finitely presented projectives. Note that

- if $S$ has one object $s$ the conclusion of the theorem is that $C$ is equivalent to the category of right $\text{End}(s)$-modules,
- if $S$ has finitely many objects $s_1, \dots s_n$ a small argument shows that the conclusion of the theorem is that $C$ is equivalent to the category of right $\text{End}(s_1 \oplus \dots \oplus s_n)$-modules, and
- if $S$ is the free $\text{Ab}$-enriched category on a quiver the conclusion of the theorem is that $C$ is equivalent to the category of (right) representations of this quiver (over $\mathbb{Z}$, but you can replace $\text{Ab}$ with vector spaces and everything goes through the same).

Applications of this theorem include Serre's affineness criterion and, I believe, the Dold-Kan theorem, although I haven't seen the details written down.

Now, there is an analogous theorem in the derived (stable $\infty$) setting which has the pleasant additional property that the projectivity hypothesis can be dropped. This makes it much easier to find full subcategories $S$ satisfying the hypotheses of the theorem; see, for example, Schwede-Shipley for a discussion in the setting of model categories. In particular, many schemes (stacks, etc.) now have the property that their derived categories of quasicoherent sheaves satisfy the hypotheses of the theorem; these are said to be "compactly generated."