Recent brain tumor classification methods often report high accuracy but rely on deep, over-parameterized architectures with limited interpretability, making it difficult to determine whether predictions are driven by tumor-relevant evidence or by spurious cues such as background artifacts or normal tissue. We propose an explainable convolutional neural network (CNN) framework that enhances model transparency without sacrificing classification accuracy. This approach supports more trustworthy AI in healthcare and contributes to SDG 3: Good Health and Well-being by enabling more dependable MRI-based brain tumor diagnosis and earlier detection. Rather than using explainable AI solely for post hoc visualization, we employ Grad-CAM to quantify layer-wise relevance and guide the removal of low-contribution layers, reducing unnecessary depth and parameters while encouraging attention to discriminative tumor regions. We further validate the model's decision rationale using complementary explainability methods, combining Grad-CAM for spatial localization with SHAP and LIME for attribution-based verification. Experiments on multi-class brain MRI datasets show that the proposed model achieves 98.21% accuracy on the primary dataset and 95.74% accuracy on an unseen dataset, indicating strong cross-dataset generalization. Overall, the proposed approach balances simplicity, transparency, and accuracy, supporting more trustworthy and clinically applicable brain tumor classification for improved health outcomes and non-invasive disease detection.