We demonstrate the formation of CdSe nanoplatelet (NPL) exciton-polaritons in a distributed bragg reflector (DBR) cavity. The molecule-cavity hybrid system is in the strong coupling regime with an 83 meV Rabi splitting, characterized from angle-resolved reflectance and photoluminescence measurements. Mixed quantum-classical dynamics simulations are used to investigate the polariton photophysics of the hybrid system by treating the electronic and photonic degrees of freedom (DOF) quantum mechanically, and the nuclear phononic DOF classically. Our numerical simulations of the angle-resolved photoluminescence (PL) agree extremely well with the experimental data, providing a fundamental explanation of the asymmetric intensity distribution of the upper and lower polariton branches. Our results also provide mechanistic insights into the importance of phonon-assisted non-adiabatic transitions among polariton states which are reflected in the various features of the PL spectra. This work proves the feasibility of coupling nanoplatelet electronic states with the photon states of a dielectric cavity to form a hybrid system and provides a new platform for investigating cavity-mediated physical and chemical processes.