Agile flight for the quadrotor cable-suspended payload system is a formidable challenge due to its underactuated, highly nonlinear, and hybrid dynamics. Traditional optimization-based methods often struggle with high computational costs and the complexities of cable mode transitions, limiting their real-time applicability and maneuverability exploitation. In this letter, we present FLARE, a reinforcement learning (RL) framework that directly learns agile navigation policy from high-fidelity simulation. Our method is validated across three designed challenging scenarios, notably outperforming a state-of-the-art optimization-based approach by a 3x speedup during gate traversal maneuvers. Furthermore, the learned policies achieve successful zero-shot sim-to-real transfer, demonstrating remarkable agility and safety in real-world experiments, running in real time on an onboard computer.