Traffic flow (TF) prediction is an important and yet a challenging task in transportation systems, since the TF involves high nonlinearities and is affected by many elements. Recently, neural networks have attracted much attention for TF prediction, but they are commonly black boxes with complex architectures and difficult to be interpreted, e.g., the contributions of specific traffic elements are not explicit, hardly providing informative guidance. In this paper, we aim at addressing more interpretable short-term TF prediction with joint consideration to high accuracy, and thus introduces a pragmatic method by applying the efficient hinging hyperplanes neural network (EHHNN) simply built upon sparse neuron connections. In the proposed method, different traffic factors are incorporated into the inputs, including their spatial-temporal information. Besides the pursuit of accuracy, we further extend the ANOVA decomposition of EHHNNs to the interpretation analysis with specifications to traffic data, in which the contributions concerning specific traffic variables are detected quantitatively. As such, the proposed method firstly applies the EHHNN to filter out more important traffic variables for dimensionality reduction while maintaining accurate prediction. Then, variable interpretation analysis is performed from different perspectives, e.g. to quantitatively investigate the influence of traffic factors and also their spatial-temporal impacts. Therefore, a predictor and an analyzing tool can both be attained for the TF by exerting the flexibility and extending the interpretability of EHHNNs, which is promising to provide informative guidance to future traffic control. Numerical experiments verify the effectiveness and potential of the proposed method in TF prediction and analysis.