Project-based learning (PBL) has been a methodology traditionally associated with student engagement and good results. However, not all experiences are sufficiently satisfactory. Comprehensive models that explain the success or failure of these experiences are still lacking. The objective of this study was to understand the mechanisms that explain student engagement and other satisfactory educational results of PBL. During two academic years, the Sustainable Urban Race (SUR) project was analyzed. In this project, students from secondary schools should design and build an electric vehicle using solar energy. In the present study, a multigroup analysis of structural equations was applied. The data showed a positive association of the challenging and support-enriched context with the students' engagement, being partially mediated by the satisfaction of the competence needs, peer relationships and group autonomy. The study provides a useful framework for practitioners and researchers of student engagement in PBL.