The construction of the equivalence is described in the articles the author cites, but these are somewhat old references so it might be helpful if I try to translate it into modern language.
It is known that $\mathrm{Coh}(\mathbb{P}^2)$ is derived equivalent to the category $\mathrm{mod}\Lambda$, where $\Lambda$ is an endomorphism algebra $\mathrm{End}(\mathcal{O}(n)\oplus\mathcal{O}(n+1))$. It is straightforward to check that this endomorphism algebra is isomorphic to $K_3$:
$$
\begin{split}
\mathrm{End}(\mathcal{O}(n)) \simeq \Gamma\mathcal{O} \simeq \mathbb C,\quad
\mathrm{Hom}(\mathcal{O}(n),\mathcal{O}(n+1)) \simeq \Gamma\mathcal{O}(1) \simeq \mathbb C^3,\\
\mathrm{End}(\mathcal{O}(n+1)) \simeq \Gamma\mathcal{O} \simeq \mathbb C,\quad
\mathrm{Hom}(\mathcal{O}(n),\mathcal{O}(n-1)) \simeq \Gamma\mathcal{O}(-1) = 0,\\
\end{split}
$$
To a sheaf $\mathcal{F}$ (more generally a complex of sheaves) one associates the Kronecker module $R\mathrm{Hom}(\mathcal{O}(n)\oplus\mathcal{O}(n+1),\mathcal{F})$, with module structure given by precomposition with endomorphisms in $\Lambda$. An isomorphism between two such modules $M,N$ of dimension vector $(a,b)$ is given by a pair of invertible matrices in $\mathrm{GL}(a)\times \mathrm{GL}(b)$, so dividing out by the action of this group defines the (coarse) moduli space.
As shown in the reference [3] of your article, any representative in the moduli space $M_{\mathbb{P}^2}(0,2)$ is determined up to an action of $\mathrm{GL}(2)\times\mathrm{GL}(2)$ by a map $$\alpha: \Gamma\mathcal{O}(1)\otimes_\mathbb{C}H^1\mathcal{F}(-2)\to H^1\mathcal{F}(-1).$$
For the sheaves considered here, such a map is equivalent to the module
$$
R\mathrm{Hom}(\mathcal{O}(1)\oplus\mathcal{O}(2),\mathcal{F}[1]) = H^1\mathcal{F}(-1)\oplus H^1\mathcal{F}(-2),
$$
where we've chosen $n=1$ (the arrows run opposite to how you may expect because the module structure is given by precomposition) Both of the cohomology spaces are of rank $2$, so after shifting by $[-1]$ this gives one a module of dimension vector $(2,2)$. A stability condition can be defined on the level of derived categories through Bridgeland stability, which will reduce to the stability on sheaves and modules on the respective sides of the equivalence. This should motivate why the semistable modules $M^{ss}_{(2,2)}(K_3,(-1,1))$ correspond to the sheaves $M_{\mathbb{P}^2}(0,2)$.
The equivalence between semi-stable modules of dimension vector $(2,2)$ and $(2,4)$ is provided by a reflection functor. Reflecting at the second vertex will yield a functor which has the following effect on the dimension vector
$$
\sigma_1: \begin{pmatrix}a\\b\end{pmatrix}\mapsto \begin{pmatrix}a\\b\end{pmatrix} - (2b-\#(0\to 1)a)\begin{pmatrix}0\\1\end{pmatrix},
$$
where $\#(0\to 1) = 3$ is the number of arrows between the vertices. The proof that the functor is in fact a derived equivalence and preserves stability relies on tilting theory. I am not the right person to recommend you any books on tilting theory, but I can say that chapter 7 of "Representations of Finite-Dimensional Algebras" by Gabriel and Roiter is helpful for understanding some classical theory on reflection for quivers.
