While deep neural networks have achieved impressive performance on a range of NLP tasks, these data-hungry models heavily rely on labeled data, which restricts their applications in scenarios where data annotation is expensive. Natural language (NL) explanations have been demonstrated very useful additional supervision, which can provide sufficient domain knowledge for generating more labeled data over new instances, while the annotation time only doubles. However, directly applying them for augmenting model learning encounters two challenges: (1) NL explanations are unstructured and inherently compositional, which asks for a modularized model to represent their semantics, (2) NL explanations often have large numbers of linguistic variants, resulting in low recall and limited generalization ability. In this paper, we propose a novel Neural Execution Tree (NExT) framework to augment training data for text classification using NL explanations. After transforming NL explanations into executable logical forms by semantic parsing, NExT generalizes different types of actions specified by the logical forms for labeling data instances, which substantially increases the coverage of each NL explanation. Experiments on two NLP tasks (relation extraction and sentiment analysis) demonstrate its superiority over baseline methods. Its extension to multi-hop question answering achieves performance gain with light annotation effort.