Large language models (LLMs) greatly affect language learning, but research on Chinese EFL (English as a Foreign Language) learners' engagement is limited. In this sense, the present study draws upon Self-Determination Theory (SDT) and employs a mixed-methods approach to explore how 210 Chinese EFL Learners engage with LLMs. Findings showed that most of the basic psychological needs (BPNs) play a critical role in predicting behavioral, cognitive, and emotional engagement. However, perceived autonomy did not emerge as a predictor for behavioral engagement, and perceived competence was not found to be a predictor for either behavioral engagement or cognitive engagement. Furthermore, our qualitative interviews showed that the influence of BPNs on EFL learners' engagement with LLMs can be categorized into six thematic areas: self-directed learning empowerment, goal-oriented learning challenges, individual performance enhancement, limited knowledge advancement, collaborative learning access, and interpersonal connection gaps. The findings of this study could provide insights for foreign language teachers in their instructional design and for policymakers in formulating relevant policies.