A detailed theoretical and experimental study is carried out to understand the effect of transition metal atom (TM) substitution at the Cu tetrahedral site in synthetic tetrahedrite material Cu$_{12-x}$TM$_x$Sb$_4$S$_{13}$ (TM = Mn, Fe, Co, Ni, and Zn). The samples are prepared by solid state synthesis method with the desired compositions. The X-ray diffraction (XRD) pattern of all the samples reveal tetrahedrite as the main phase with traces of secondary phases, confirmed by Electron Probe Micro Analysis (EPMA). X-ray Photoelectron Spectroscopy (XPS) reveals that TM is in +2 oxidation state for all samples except for Fe which shows a +3 oxidation state. Ultraviolet Photoelectron Spectroscopy (UPS) measurements show the highest work function for Cu$_{11.5}$Co$_{0.5}$Sb$_4$S$_{13}$, indicating high band degeneracy. Density Functional Theory (DFT) calculations reveal that TM substitution introduces spin polarized states within the band structure, thus, changing the band degeneracy and density of states (DOS) at the Fermi level (E$_F$). It is confirmed from DFT calculations that band degeneracy is highest for Cu$_{11.5}$Co$_{0.5}$Sb$_4$S$_{13}$ among the TM substituted samples, with high DOS at E$_F$, which is experimentally confirmed by magnetic susceptibility analysis. The electrical resistivity (ρ) and Seebeck coefficients (S) of the substituted samples are higher than the pristine compound due to compensation of holes caused by substitution of TM$^{+2}$ or TM$^{+3}$ on the Cu tetrahedral site. Because of the high band degeneracy and DOS at E$_F$, a high power factor is achieved for the composition Cu$_{11.5}$Co$_{0.5}$Sb$_4$S$_{13}$, enabling it to attain the maximum figure of merit (zT) among the substituted tetrahedrite compositions. The paper presents a comprehensive study to understand the role of magnetic impurities (TM) in influencing the band structure and, hence, the transport properties in substituted tetrahedrite.