Traditional equation-driven hydrological models often struggle to accurately predict streamflow in challenging regional Earth systems like the Tibetan Plateau, while hybrid and existing algorithm-driven models face difficulties in interpreting hydrological behaviors. This work introduces HydroTrace, an algorithm-driven, data-agnostic model that substantially outperforms these approaches, achieving a Nash-Sutcliffe Efficiency of 98% and demonstrating strong generalization on unseen data. Moreover, HydroTrace leverages advanced attention mechanisms to capture spatial-temporal variations and feature-specific impacts, enabling the quantification and spatial resolution of streamflow partitioning as well as the interpretation of hydrological behaviors such as glacier-snow-streamflow interactions and monsoon dynamics. Additionally, a large language model (LLM)-based application allows users to easily understand and apply HydroTrace's insights for practical purposes. These advancements position HydroTrace as a transformative tool in hydrological and broader Earth system modeling, offering enhanced prediction accuracy and interpretability.