[en] We present first-time measurements of the Fermi surface and low-energy electronic structure of intermetallic compounds Gd₂PdSi₃ and Tb₂PdSi₃ by means of angle-resolved photoelectron spectroscopy (ARPES). We show that the Fermi surface in both compounds consists of an electron barrel at the Γ point surrounded by spindle-shaped electron pockets originating from the same band, with the band bottom of both features lying at 0.5 eV below the Fermi level. From the experimentally measured band structure, we estimate the momentum-dependent RKKY coupling strength and demonstrate that it is peaked at the 1/2Γ K wave vector. Comparison with neutron diffraction data from the same crystals shows perfect agreement of this vector with the propagation vector of the low-temperature in-plane magnetic order, thereby demonstrating the decisive role of the Fermi surface geometry in explaining the complex magnetically ordered ground state of ternary rare earth silicides.