Adversarial attacks have demonstrated the vulnerability of Machine Learning (ML) image classifiers in Synthetic Aperture Radar (SAR) Automatic Target Recognition (ATR) systems. An adversarial attack can deceive the classifier into making incorrect predictions by perturbing the input SAR images, for example, with a few scatterers attached to the on-ground objects. Therefore, it is critical to develop robust SAR ATR systems that can detect potential adversarial attacks by leveraging the inherent uncertainty in ML classifiers, thereby effectively alerting human decision-makers. In this paper, we propose a novel uncertainty-aware SAR ATR for detecting adversarial attacks. Specifically, we leverage the capability of Bayesian Neural Networks (BNNs) in performing image classification with quantified epistemic uncertainty to measure the confidence for each input SAR image. By evaluating the uncertainty, our method alerts when the input SAR image is likely to be adversarially generated. Simultaneously, we also generate visual explanations that reveal the specific regions in the SAR image where the adversarial scatterers are likely to to be present, thus aiding human decision-making with hints of evidence of adversarial attacks. Experiments on the MSTAR dataset demonstrate that our approach can identify over 80% adversarial SAR images with fewer than 20% false alarms, and our visual explanations can identify up to over 90% of scatterers in an adversarial SAR image.