Preventing collisions in multi-robot navigation is crucial for deployment. This requirement hinders the use of learning-based approaches, such as multi-agent reinforcement learning (MARL), on their own due to their lack of safety guarantees. Traditional control methods, such as reachability and control barrier functions, can provide rigorous safety guarantees when interactions are limited only to a small number of robots. However, conflicts between the constraints faced by different agents pose a challenge to safe multi-agent coordination. To overcome this challenge, we propose a method that integrates multiple layers of safety by combining MARL with safety filters. First, MARL is used to learn strategies that minimize multiple agent interactions, where multiple indicates more than two. Particularly, we focus on interactions likely to result in conflicting constraints within the engagement distance. Next, for agents that enter the engagement distance, we prioritize pairs requiring the most urgent corrective actions. Finally, a dedicated safety filter provides tactical corrective actions to resolve these conflicts. Crucially, the design decisions for all layers of this framework are grounded in reachability analysis and a control barrier-value function-based filtering mechanism. We validate our Layered Safe MARL framework in 1) hardware experiments using Crazyflie drones and 2) high-density advanced aerial mobility (AAM) operation scenarios, where agents navigate to designated waypoints while avoiding collisions. The results show that our method significantly reduces conflict while maintaining safety without sacrificing much efficiency (i.e., shorter travel time and distance) compared to baselines that do not incorporate layered safety. The project website is available at https://dinamo-mit.github.io/Layered-Safe-MARL/