Glaucoma is an increasing ocular disease and one of the major causes of irreversible blindness globally. Thus, it is of utmost importance to diagnose glaucoma accurately and in a timely manner for appropriate clinical intervention. Retinal fundus image analysis is a non-invasive approach for glaucoma screening. However, it is a tedious and subjective process for accurate glaucoma detection. In this work, a customized convolutional neural network (CNN) model is proposed for glaucoma classification using fundus images, and an attention mechanism is employed for improved discriminative feature learning. Experiments were carried out on two publicly available datasets, namely Drishti-GS1 and ACRIMA, for unbalanced and balanced datasets, respectively. Data augmentation and hyperparameter tuning were employed for improved model generalization. To increase model explainability and trust for accurate glaucoma screening, gradient-weighted class activation mapping (Grad-CAM) is employed for accurate glaucoma screening. For accurate glaucoma screening, the proposed model attained 90.32% and 96.45% accuracy on the Drishti-GS1 and ACRIMA datasets, respectively, with higher sensitivity and AUC scores. Thus, it is evident that the proposed model employing attention mechanisms and explainable AI attained higher accuracy for accurate glaucoma screening.