Phonon polaritons (PhPs), light coupled to lattice vibrations, in the highly anisotropic polar layered material molybdenum trioxide (α-MoO₃ ) are currently the focus of intense research efforts due to their extreme subwavelength field confinement, directional propagation, and unprecedented low losses. Nevertheless, prior research has primarily concentrated on exploiting the squeezing and steering capabilities of α-MoO₃ PhPs, without inquiring much into the dominant microscopic mechanism that determines their long lifetimes, which is key for their implementation in nanophotonic applications. This study delves into the fundamental processes that govern PhP damping in α-MoO₃ by combining ab initio calculations with scattering-type scanning near-field optical microscopy (s-SNOM) and Fourier transform infrared (FTIR) spectroscopy measurements across a broad temperature range (8-300 K). The remarkable agreement between our theoretical predictions and experimental observations allows us to identify third-order anharmonic phonon-phonon scattering as the main damping mechanism of α-MoO₃ PhPs. These findings shed light on the fundamental limits of low-loss PhPs, which is a crucial factor for assessing their implementation into nanophotonic devices.