Layered biaxial hyperbolic molybdenum trioxide (α-MoO3) with weak van der Waals (vdW) interlayer bonding recently received extensive attention due to its anisotropic dielectric response to infrared (IR) radiation, which couples to the lattice vibrations and allows for manipulating the radiative energy transport. However, the understanding of IR-active phonon modes and dielectric function of it has not yet been fully achieved. Here, by utilizing mode-level first-principles analysis based on density functional theory (DFT), the phonon modes contributing to the IR dielectric response of α-MoO3 are fully determined. The anisotropic IR-active modes are identified from lattice vibration analysis, allowing for a clear evaluation of the IR absorption contribution from the weak or strong IR phonon modes. By further employing anharmonic-lattice dynamics calculations, the damping of the corresponding IR modes is directly obtained. This approach enables predictions of IR optical properties without any fitting or assumed parameters. Our predictions bridge the scientific gap of comprehensively understanding the unreported IR-active phonon modes of α-MoO3 and overall agree well with available experimental data, placing our DFT-based method at a privileged stage for accurately predicting the IR optical properties of α-MoO3. These comprehensive understandings of the IR phonons and dielectric properties of α-MoO3 pave the way for nanophotonic devices with tunable functionalities and enable design of α-MoO3 for advanced optical devices.