Trip recommendation aims to generate a sequence of points of interest (POIs) under a user's query input. Existing data-driven methods mainly fall into two categories: supervised approaches and self-supervised approaches. The former cannot fully capture the transition patterns among POIs, while the latter fail to comprehensively model user's query intents. Fortunately, privileged knowledge distillation (PKD) provides us an unique opportunity to align user's query intents with its corresponding trip in historical data. However, such knowledge alignment is implicit, which may not directly reflect the query intents. To this end, in this paper, we propose EKD-Trip, an explicit intent-enhanced knowledge distillation framework. EKD-Trip first trains a trajectory encoder (teacher model) and a trip generator jointly in a self-supervised manner. Then, a query encoder (student model) is trained via multi-task learning to extract implicit knowledge by PKD from teacher and explicit knowledge from an auxiliary task, respectively. At inference time, we use the query encoder and the trip generator to recommend trips. Extensive experiments on four real-world datasets demonstrate that EKD-Trip outperforms all baselines over three metrics, with a particularly notable improvement of 13.70% in pairs-F1.