The intelligent transportation system (ITS) is one of the key components of future transportation. Vehicle trajectory prediction, which is made possible by recent advancement in sensor and communication technology, can help ITS to improve efficiency and safety of transportation by detecting potential conflicts and accidents. However, prior works mainly focus on trajectory prediction for autonomous vehicles in small-scale scenarios. In addition, interpretability of trajectory prediction, which facilitates trustworthiness of ITS, has received scant attention in the research literature. This study proposes intention-aware interactive transformer (IIT) model to address the problem of real-time vehicle trajectory prediction in large-scale dense traffic scenarios. IIT follows the transformer architecture to process time-series data and adopts an intention-based scheme to predict future trajectories. Unlike prior works that use complicated data structure to represent the spatial relation between vehicles, what is unique about IIT is that the interactions between vehicles are captured using a multi-head attention (MHA) mechanism. Moreover, the interpretability of IIT is illustrated in two levels: inter-vehicle attention and intra-vehicle intention. Experimental results suggest that MHA significantly reduces the computational cost of data preprocessing for IIT. Consequently, IIT runs up to eight times faster than baseline models in large-scale dense traffic scenarios while only suffering an average of 7.6% accuracy degradation in a 5-s prediction horizon. Further, interpretability analysis is conducted, which reveals that the interpretability of IIT is beneficial to ITS in many ways, such as detecting potential congestion and solving driver conflicts.