We present the first sentence simplification model that learns explicit edit operations (ADD, DELETE, and KEEP) via a neural programmer-interpreter approach. Most current neural sentence simplification systems are variants of sequence-to-sequence models adopted from machine translation. These methods learn to simplify sentences as a byproduct of the fact that they are trained on complex-simple sentence pairs. By contrast, our neural programmer-interpreter is directly trained to predict explicit edit operations on targeted parts of the input sentence, resembling the way that humans might perform simplification and revision. Our model outperforms previous state-of-the-art neural sentence simplification models (without external knowledge) by large margins on three benchmark text simplification corpora in terms of SARI (+0.95 WikiLarge, +1.89 WikiSmall, +1.41 Newsela), and is judged by humans to produce overall better and simpler output sentences.