This paper proposes a framework that facilitates the energy exchange between networked microgrids (MGs) in the electricity market. An alternating direction method of multipliers (ADMM)-based robust optimization algorithm is proposed to derive the optimal energy exchange strategy for the networked MGs considering the uncertainties of the electrical load, intermittent generation, and electricity prices in the external market. The proposed method naturally lends itself to a classical market-clearing problem between two hierarchical levels comprising (i) main-grid-to-MG and (ii) MG-to-MG, aiding in the result interpretation and practical realization. Leveraging from the decentralized organization, the operational autonomy and information privacy of each MG is protected. The proposed framework is tested on a modified 144-node network with 3 MGs. The numerical experiments demonstrate the convergence of the proposed market clearing to the market equilibrium in different grid operational scenarios with different conservatism parameter settings for MG operators.