We investigate InxCd1-xO materials, where x = 0.0, 0.031, 0.063 and 0.125, to understand their high electrical conductivity and optical transparency windows, using the full-potential linearized augmented plane wave (FLAPW) method. In addition, we employ the screened exchange LDA (sX-LDA) method to evaluate accurate band structures including band gap that is underestimated by the LDA calculations. The results show a dramatic Burstein-Moss shift of the absorption edge by the In doping, reflecting the small effective mass of the Cd 5s conduction band. The calculated direct band gaps, 2.36 eV for x=0.0 and 3.17 eV for x= 0.063, show excellent agreement with experiment. The effective mass of the conduction band of CdO is calculated to be 0.24 in, (in the Delta direction), in good agreement with an experimental value of 0.27m(e), explaining its high electrical conductivity. The hybridization between the Cd 5s and the In 5s states yields complex many-body effects in the conduction bands: a hybridization gap in the conduction bands and a band-gap narrowing which cancels the further Burstein-Moss shift for higher In doping.