Autonomous Driving (AD) vehicles still struggle to exhibit human-like behavior in highly dynamic and interactive traffic scenarios. The key challenge lies in AD's limited ability to interact with surrounding vehicles, largely due to a lack of understanding the underlying mechanisms of social interaction. To address this issue, we introduce MPCFormer, an explainable socially-aware autonomous driving approach with physics-informed and data-driven coupled social interaction dynamics. In this model, the dynamics are formulated into a discrete space-state representation, which embeds physics priors to enhance modeling explainability. The dynamics coefficients are learned from naturalistic driving data via a Transformer-based encoder-decoder architecture. To the best of our knowledge, MPCFormer is the first approach to explicitly model the dynamics of multi-vehicle social interactions. The learned social interaction dynamics enable the planner to generate manifold, human-like behaviors when interacting with surrounding traffic. By leveraging the MPC framework, the approach mitigates the potential safety risks typically associated with purely learning-based methods. Open-looped evaluation on NGSIM dataset demonstrates that MPCFormer achieves superior social interaction awareness, yielding the lowest trajectory prediction errors compared with other state-of-the-art approaches. The prediction achieves an ADE as low as 0.86 m over a long prediction horizon of 5 seconds. Close-looped experiments in highly intense interaction scenarios, where consecutive lane changes are required to exit an off-ramp, further validate the effectiveness of MPCFormer. Results show that MPCFormer achieves the highest planning success rate of 94.67%, improves driving efficiency by 15.75%, and reduces the collision rate from 21.25% to 0.5%, outperforming a frontier Reinforcement Learning (RL) based planner.