Anomaly detection is one of the most significant tasks in multivariate time series analysis, while it remains challenging to model complex patterns for improving detection accuracy and to interpret the root causes of anomalies. However, existing studies either consider only the temporal dependencies, or simply reconstruct the original input for detection, both neglecting the hidden relationships among multivariate. We propose an adversarial graph neural network based anomaly detection model, called SGAT-AE, which consists of a Self-learning Graph ATtention network (SGAT), an Auto-Encoder (AE), and an adversarial training component. Specifically, SGAT is a prediction model that discovers the graph dependency relationships among multivariate and acts as a sample generator to confuse AE, while AE reconstructs the samples and acts as a discriminator that distinguishes a real sample from a generated one. A novel adversarial training between SGAT and AE is applied to amplify the errors of anomalies such that the prediction performance of SGAT is improved and the overfitting of AE is avoided. In addition, we aggregate the prediction error, the reconstruction error, and the adversarial error for anomaly detection, and develop a graph based anomaly interpretation method that locates the root causes from both local and global perspectives. Extensive experiments with five real-world data offer evidence that the proposed solution SGAT-AE is capable of achieving better performance when compared with the state-of-the-art proposals.