Let G be a compact Lie group of ADE type. Compactifying the N=(2,0) 6d SCFT of type G on a 3-manifold M3 gives a N=2 3d QFT TG[M3]. More precisely, to have SUSY in 3d, one needs to make a topological twist along M3, which is the one obtained from an embedding M3⊂T∗M3 in a M-theory realization.
The claim is that the moduli space of SUSY vacua of TG[M3] on R2×S1 is Mflat(M3,GC), the moduli space of flat connections on M3 for the complexified gauge group GC. 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 S1. One then gets at low energy maximally, i.e. 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 M3, 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 M3, which can be naturally combined with the gauge field on M3 to obtained a complexified gauge field on M3. Supersymmetry requires this compexified gauge field on M3 to be flat.