With the rapid growth of scholarly archives, researchers subscribe to "paper alert" systems that periodically provide them with recommendations of recently published papers that are similar to previously collected papers. However, researchers sometimes struggle to make sense of nuanced connections between recommended papers and their own research context, as existing systems only present paper titles and abstracts. To help researchers spot these connections, we present PaperWeaver, an enriched paper alerts system that provides contextualized text descriptions of recommended papers based on user-collected papers. PaperWeaver employs a computational method based on Large Language Models (LLMs) to infer users' research interests from their collected papers, extract context-specific aspects of papers, and compare recommended and collected papers on these aspects. Our user study (N=15) showed that participants using PaperWeaver were able to better understand the relevance of recommended papers and triage them more confidently when compared to a baseline that presented the related work sections from recommended papers.