A half-Heusler material FeNb$_{0.8}$Ti$_{0.2}$Sb has been identified as a promising thermoelectric material due to its excellent thermoelectric performance at high temperatures. The origins of the efficient thermoelectric performance are investigated through a series of low-temperature (2 - 400 K) measurements. The high data coherence of the low and high temperatures is observed. An optimal and nearly temperature-independent carrier concentration is identified, which is ideal for the power factor. The obtained single type of hole carrier is also beneficial to the large Seebeck coefficient. The electronic thermal conductivity is found to be comparable to the lattice thermal conductivity and becomes the dominant component above 200 K. These findings again indicate that electron scattering plays a key role in the electrical and thermal transport properties. The dimensionless figure of merit is thus mainly governed by the electronic properties. These effects obtained at low temperatures with the avoidance of possible thermal fluctuations together offer the physical origin for the excellent thermoelectric performance in this material.