One-dimensional particle chains are fundamental models to explain anomalous thermal conduction in low-dimensional solids such as nanotubes and nanowires. In these systems the thermal energy is carried by phonons, i.e., propagating lattice oscillations that interact via nonlinear resonance. The average energy transfer between the phonons can be described by the wave kinetic equation, derived directly from the microscopic dynamics. Here we use the spatially nonhomogeneous wave kinetic equation of the prototypical β-Fermi-Pasta-Ulam-Tsingou model, to study thermal conduction in one-dimensional particle chains on a mesoscale description. By means of numerical simulations, we study two complementary aspects of thermal conduction: in the presence of thermostats setting different temperatures at the two ends and propagation of a temperature perturbation over an equilibrium background. Our main findings are as follows. (i) The anomalous scaling of the conductivity with the system size, in close agreement with the known results from the microscopic dynamics, is due to a nontrivial interplay between high and low wave numbers. (ii) The high-wave-number phonons relax to local thermodynamic equilibrium transporting energy diffusively, in the manner of Fourier. (iii) The low-wave-number phonons are nearly noninteracting and transfer energy ballistically. These results present perspectives for the applicability of the full nonhomogeneous wave kinetic equation to study thermal propagation.