Modern recommender systems use ML models to predict consumer preferences from consumption history. Although these "black-box" models achieve impressive predictive performance, they often suffer from a lack of transparency and explainability. Contrary to the presumed tradeoff between explainability and accuracy, we show that integrating large language models (LLMs) with deep neural networks (DNNs) can improve both. We propose LR-Recsys, which augments DNN-based systems with LLM reasoning capabilities. LR-Recsys introduces a contrastive-explanation generator that produces human-readable positive explanations and negative explanations. These explanations are embedded via a fine-tuned autoencoder and combined with consumer and product features to improve predictions. Beyond offering explainability, we show that LR-Recsys also improves learning efficiency and predictive accuracy, as supported by high-dimensional, multi-environment statistical learning theory. LR-Recsys outperforms state-of-the-art recommender systems by 3-14% on three real-world datasets. Importantly, our analysis reveals that these gains primarily derive from LLMs' reasoning capabilities rather than their external domain knowledge. LR-RecSys presents an effective approach to combine LLMs with traditional DNNs, two of the most widely used ML models today. The explanations generated by LR-Recsys provide actionable insights for consumers, sellers, and platforms, helping to build trust, optimize product offerings, and inform targeting strategies.