Airport taxi delays adversely affect airports and airlines around the world in terms of congestion, operational workload, and environmental emissions. Departure Metering (DM) is a promising approach to contain taxi delays by controlling departure pushback times. The key idea behind DM is to transfer aircraft waiting time from taxiways to gates. State-of-the-art DM methods use model-based control policies that rely on airside departure modeling to obtain simplified analytical equations. Consequently, these models fail to capture non-stationarity existing in the complex airside operations and the policies perform poorly under uncertainties. In this work, we propose model-free and learning-based DM using Deep Reinforcement Learning (DRL) approach to reduce taxi delays while meeting flight schedule constraints. We cast the DM problem in an MDP framework and develop a representative airport simulator to simulate airside operations and evaluate the learnt DM policy. For effective state representation, we introduce features to capture both local and airport-wide congestion levels. Finally, the performance of multiple agentssharing the same trained policy, is evaluated on different traffic densities. The proposed approach shows a reduction of up to 25% in taxi delays in medium traffic scenarios. Moreover, upon experiencing increased traffic density, taxi time savings achieved by proposed algorithm significantly increase while the average gate holding times do not increase as much. Results demonstrate that DRL can learn an effective DM policy to better manage airside traffic and contain congestion on the taxiways.