Cu₂SnSe₃ exhibits exceptionally low lattice thermal conductivity (κ_L) and a nonclassical temperature dependence among ternary copper-based diamondoid compounds, yet the microscopic origins of these phonon behaviors remain poorly understood. In this work, we theoretically investigate the intrinsic phonon transport in Cu₂SnSe₃ by employing a two-channel thermal transport model that separates particle-like (κ_p) and coherent contributions (κ_c), while fully incorporating both three-phonon (3ph) and four-phonon (4ph) scattering processes. Our calculations show that κ_c is negligible at 300 K but becomes progressively more significant at elevated temperatures, while κ_p remains the dominant contributor to κ_L over the entire temperature range. 4ph scattering is identified as the primary factor driving the deviation from the classical T^-1 dependence, yielding a temperature scaling of T^-1.1 that closely matches the experimental T^-1.2 behavior. This strong 4ph scattering originates from flat optical phonon modes in the 1.2-2.2 THz range, which provide an exceptionally large scattering phase space. The low κ_L (e.g., ~ 0.8 W m^-1 K^-1 at 773 K) of Cu₂SnSe₃ relative to other ternary copper-based diamondoid compounds is attributed to the asymmetric potential energy of Cu atoms, which induces large atomic displacement parameters and pronounced anharmonicity in the Cu-related low-frequency optical branches. Furthermore, qualitative analyses of cation disorder and anion substitution show that disorder leads to significant phonon broadening, while replacing Se with S hardens the acoustic branches and markedly shifts the optical modes upward. Our findings not only clarify the origins of the nonclassical temperature dependence and low κ_L in Cu₂SnSe₃, but also provide a general guideline for designing diamondoid thermoelectrics with intrinsically suppressed κ_L.