In real-world maintenance applications, deep generative models have shown promising performance in detecting anomalous events of entities from time-series signals collected from multiple sensors. Nevertheless, we outline two important challenges of leveraging such models for times-series anomaly detection: 1) developing effective and efficient reconstruction models and 2) exploiting the similarity and interrelation structures among the multivariate time series data channels. To address these challenges, in this paper we propose a stacking variational auto-encoder (VAE) model with graph neural networks for the effective and interpretable time-series anomaly detection. Specifically, we propose a stacking block-wise reconstruction framework with a weight-sharing scheme for the multivariate time series data with similarities among channels. Moreover, with a graph learning module, our model learns a sparse adjacency matrix to explicitly capture the stable interrelation structure information among multiple time series data channels for interpretable reconstruction of series patterns. Experimental results show that our proposed model outperforms the strong baselines on three public datasets with considerable improvements and meanwhile still maintains the training efficiency. Furthermore, we demonstrate that the intuitive stable structure learned by our model significantly improves the interpretability of our detection results.