Student feedback engagement is often reported as unsatisfactorily low, undermining the effectiveness of feedback in higher education. The emergence of Generative Artificial Intelligence (GenAI) has created new opportunities to enhance student feedback engagement through its unique affordances. However, assuming that GenAI is a panacea for feedback disengagement can be misleading, as it also presents substantial challenges. Therefore, it is essential to understand how students engage comprehensively with GenAI feedback. However, such understanding is lacking in literature. To close the research gaps, this study investigated how college students engaged with the feedback provided by ChatGPT on their International English Language Testing System writing tasks. Sixteen undergraduates voluntarily participated in unobtrusive observations and stimulated recall interviews. Their feedback engagement was classified into cognitive, metacognitive, affective and behavioural aspects. The findings reveal that participants employed more cognitive strategies than metacognitive ones, experienced minimal emotional resistance, and exhibited varying levels of trust in ChatGPT's feedback. Additionally, the participants' behavioural engagement appeared superficial when requesting feedback, interacting with ChatGPT and revising their writing tasks. Student feedback literacy, including prompt engineering, evaluative judgement, emotional reflexivity, ethical decision-making and meta-cognitive skills, should be cultivated to foster students' active engagement in a GenAI context.