With the rapid development of mobile-internet technologies, on-demand ride-sourcing services have become increasingly popular and largely reshaped the way people travel. Demand prediction is one of the most fundamental components in supply-demand management systems of ride-sourcing platforms. With accurate short-term prediction for origin-destination (OD) demand, the platforms make precise and timely decisions on real-time matching, idle vehicle reallocations and ride-sharing vehicle routing, etc. Compared to zone-based demand prediction that has been examined by many previous studies, OD-based demand prediction is more challenging. This is mainly due to the complicated spatial and temporal dependencies among demand of different OD pairs. To overcome this challenge, we propose the Spatio-Temporal Encoder-Decoder Residual Multi-Graph Convolutional network (ST-ED-RMGC), a novel deep learning model for predicting ride-sourcing demand of various OD pairs. Firstly, the model constructs OD graphs, which utilize adjacent matrices to characterize the non-Euclidean pair-wise geographical and semantic correlations among different OD pairs. Secondly, based on the constructed graphs, a residual multi-graph convolutional (RMGC) network is designed to encode the contextual-aware spatial dependencies, and a long-short term memory (LSTM) network is used to encode the temporal dependencies, into a dense vector space. Finally, we reuse the RMGC networks to decode the compressed vector back to OD graphs and predict the future OD demand. Through extensive experiments on the for-hire-vehicles datasets in Manhattan, New York City, we show that our proposed deep learning framework outperforms the state-of-arts by a significant margin.