The end-to-end (E2E) paradigm, which maps sensor inputs directly to driving decisions, has recently attracted significant attention due to its unified modeling capability and scalability. However, ensuring safety in this unified framework remains one of the most critical challenges. In this work, we propose SafeDrive, an E2E planning framework designed to perform explicit and interpretable safety reasoning through a trajectory-conditioned Sparse World Model. SafeDrive comprises two complementary networks: the Sparse World Network (SWNet) and the Fine-grained Reasoning Network (FRNet). SWNet constructs trajectory-conditioned sparse worlds that simulate the future behaviors of critical dynamic agents and road entities, providing interaction-centric representations for downstream reasoning. FRNet then evaluates agent-specific collision risks and temporal adherence to drivable regions, enabling precise identification of safety-critical events across future timesteps. SafeDrive achieves state-of-the-art performance on both open-loop and closed-loop benchmarks. On NAVSIM, it records a PDMS of 91.6 and an EPDMS of 87.5, with only 61 collisions out of 12,146 scenarios (0.5%). On Bench2Drive, SafeDrive attains a 66.8% driving score.