Deep learning models based on resting-state functional magnetic resonance imaging (rs-fMRI) have been widely used to diagnose brain diseases, particularly autism spectrum disorder (ASD). Existing studies have leveraged the functional connectivity (FC) of rs-fMRI, achieving notable classification performance. However, they have significant limitations, including the lack of adequate information while using linear low-order FC as inputs to the model, not considering individual characteristics (i.e., different symptoms or varying stages of severity) among patients with ASD, and the non-explainability of the decision process. To cover these limitations, we propose a novel explainability-guided region of interest (ROI) selection (EAG-RS) framework that identifies non-linear high-order functional associations among brain regions by leveraging an explainable artificial intelligence technique and selects class-discriminative regions for brain disease identification. The proposed framework includes three steps: (i) inter-regional relation learning to estimate non-linear relations through random seed-based network masking, (ii) explainable connection-wise relevance score estimation to explore high-order relations between functional connections, and (iii) non-linear high-order FC-based diagnosis-informative ROI selection and classifier learning to identify ASD. We validated the effectiveness of our proposed method by conducting experiments using the Autism Brain Imaging Database Exchange (ABIDE) dataset, demonstrating that the proposed method outperforms other comparative methods in terms of various evaluation metrics. Furthermore, we qualitatively analyzed the selected ROIs and identified ASD subtypes linked to previous neuroscientific studies.