In this paper, we report on terahertz frequency-domain spectroscopy (THz-FDS) experiments in which we measure charge carrier dynamics and excitations of thin-film superconducting systems at low temperatures in the THz spectral range. The characteristics of the set-up and the experimental procedures are described comprehensively. We discuss the single-particle density of states and a theory of electrodynamic absorption and optical conductivity of conventional superconductors. We present the experimental performance of the setup at low temperatures for a broad spectral range from 3 to 38 cm-1(0.1-1.1 THz) by the example of ultra-thin films of weakly disordered superconductors niobium nitride (NbN) and tantalum nitride (TaN) with different values of critical temperaturesTc. Furthermore, we analyze and interpret our experimental data within the framework of conventional Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. By and large, we find the properties of our NbN and TaN thin films to be well described by the theory. Our results on NbN resemble tendencies towards anomalous behavior of the ratio 2Δ(0)/kBTcas a function ofTc.