Let $G$ be a compact Lie group of ADE type. Compactifying the $\mathcal{N}=(2,0)$ 6d SCFT of type $G$ on a 3-manifold $M_3$ gives a $\mathcal{N}=2$ 3d QFT $T_G[M_3]$. More precisely, to have SUSY in 3d, one needs to make a topological twist along $M_3$, which is the one obtained from an embedding $M_3 \subset T^{*}M_3$ in a M-theory realization.

The claim is that the moduli space of SUSY vacua of $T_G[M_3]$ on $\mathbb{R}^2 \times S^1$ is $M_{flat}(M_3,G_{\mathbb{C}})$, the moduli space of flat connections on $M_3$ for the complexified gauge group $G_{\mathbb{C}}$. See for example section 5 of

https://arxiv.org/abs/1108.4389

To understand this fact, it is better to start in 6d and to first compactify on $S^1$. One then gets at low energy maximally, i.e. $\mathcal{N}=2$ , 5d super-Yang-Mills theory of group $G$. The advantage is that this 5d theory has an explicit Lagrangian description, which is not the case for the 6d theory. One can then compactify 5d super-Yang-Mills on $M_3$, with appropriate topological twist. In 5d super-Yang-Mills on flat space, one has 5 real scalar fields in the adjoint representation. After the topological twist, 3 of them becomes 1-forms on $M_3$, which can be naturally combined with the gauge field on $M_3$ to obtained a complexified gauge field on $M_3$. Supersymmetry requires this compexified gauge field on $M_3$ to be flat.