The abrupt transition to Emergency Remote Learning during the COVID-19 pandemic significantly elevated psychological stress among higher education students. Yet, most existing research studies have either identified factors contributed by COVID-19 as stress predictors or general stress perceived as a result of the pandemic. Moreover, these studies predominantly employ binary stress detection using generic psychometric instruments, excluding stressors related to Emergency Remote Learning as a pedagogical shift, thereby limiting insights for targeted interventions. This study addresses these gaps by developing a comprehensive, multiclass stress classification framework using a novel questionnaire dataset on perceived stress, which captures multidimensional Psychosocial, Environmental, Pedagogical, and Demographic stressors. Eight different Machine learning models were systematically evaluated through nested cross-validation with hyperparameter tuning. Logistic Regression achieved superior performance (93.67±1.09% accuracy, MCC=0.915), outperforming complex ensembles with healthy train-test generalization. Multiple complementary Explainable Artificial Intelligence techniques consistently highlighted specific features as dominant predictors of elevated stress during Emergency Remote Learning, surpassing traditional Stress scales. Key contributors to severe stress included reduced peer interaction, emotional disconnect between peers and instructors, whereas personal agency and coping-related items demonstrated protective effects. By integrating multiclass prediction with robust interpretability, this study provides a transparent framework for identifying heterogeneous stress levels among students, a capability largely absent in existing literature and previously unexplored in the Indian context. The findings advance precision risk stratification and can guide the design of targeted psychosocial and pedagogical interventions for remote and hybrid learning environments.