Much of modern-day text simplification research focuses on sentence-level simplification, transforming original, more complex sentences into simplified versions. However, adding content can often be useful when difficult concepts and reasoning need to be explained. In this work, we present the first data-driven study of content addition in text simplification, which we call elaborative simplification. We introduce a new annotated dataset of 1.3K instances of elaborative simplification in the Newsela corpus, and analyze how entities, ideas, and concepts are elaborated through the lens of contextual specificity. We establish baselines for elaboration generation using large-scale pre-trained language models, and demonstrate that considering contextual specificity during generation can improve performance. Our results illustrate the complexities of elaborative simplification, suggesting many interesting directions for future work.