In many regions globally, snowmelt-recharged mountainous karst aquifers serve as crucial sources for municipal and agricultural water supplies. In these watersheds, complex interplay of meteorological, topographical, and hydrogeological factors leads to intricate recharge-discharge pathways. This study introduces a spatially distributed deep learning precipitation-runoff model that combines Convolutional Long Short-Term Memory (ConvLSTM) with a spatial attention mechanism. The effectiveness of the deep learning model was evaluated using data from the Logan River watershed and subwatersheds, a characteristically karst-dominated hydrological system in northern Utah. Compared to the ConvLSTM baseline, the inclusion of a spatial attention mechanism improved performance for simulating discharge at the watershed outlet. Analysis of attention weights in the trained model unveiled distinct areas contributing the most to discharge under snowmelt and recession conditions. Furthermore, fine-tuning the model at subwatershed scales provided insights into cross-subwatershed subsurface connectivity. These findings align with results obtained from detailed hydrogeochemical tracer studies. Results highlight the potential of the proposed deep learning approach to unravel the complexities of karst aquifer systems, offering valuable insights for water resource management under future climate conditions. Furthermore, results suggest that the proposed explainable, spatially distributed, deep learning approach to hydrologic modeling holds promise for non-karstic watersheds.