Accurately determining the shape and location of internal structures within deformable objects is crucial for medical tasks that require precise targeting, such as robotic biopsies. We introduce LUDO, a method for accurate low-latency understanding of deformable objects. LUDO reconstructs objects in their deformed state, including their internal structures, from a single-view point cloud observation in under 30 ms using occupancy networks. We demonstrate LUDO's abilities for autonomous targeting of internal regions of interest (ROIs) in highly deformable objects. Additionally, LUDO provides uncertainty estimates and explainability for its predictions, both of which are important in safety-critical applications such as surgical interventions. We evaluate LUDO in real-world robotic experiments, achieving a success rate of 98.9% for puncturing various ROIs inside highly deformable objects. LUDO demonstrates the potential to interact with deformable objects without the need for deformable registration methods.