Reconstructing high-resolution rainfall fields is essential for flood forecasting, hydrological modeling, and climate analysis. However, existing spatial interpolation methods-whether based on automatic weather station (AWS) measurements or enhanced with satellite/radar observations often over-smooth critical structures, failing to capture sharp transitions and localized extremes. We introduce RainSeer, a structure-aware reconstruction framework that reinterprets radar reflectivity as a physically grounded structural prior-capturing when, where, and how rain develops. This shift, however, introduces two fundamental challenges: (i) translating high-resolution volumetric radar fields into sparse point-wise rainfall observations, and (ii) bridging the physical disconnect between aloft hydro-meteors and ground-level precipitation. RainSeer addresses these through a physics-informed two-stage architecture: a Structure-to-Point Mapper performs spatial alignment by projecting mesoscale radar structures into localized ground-level rainfall, through a bidirectional mapping, and a Geo-Aware Rain Decoder captures the semantic transformation of hydro-meteors through descent, melting, and evaporation via a causal spatiotemporal attention mechanism. We evaluate RainSeer on two public datasets-RAIN-F (Korea, 2017-2019) and MeteoNet (France, 2016-2018)-and observe consistent improvements over state-of-the-art baselines, reducing MAE by over 13.31% and significantly enhancing structural fidelity in reconstructed rainfall fields.