We introduce MapStory, an LLM-powered animation authoring tool that generates editable map animation sequences directly from natural language text. Given a user-written script, MapStory leverages an agentic architecture to automatically produce a scene breakdown, which decomposes the script into key animation building blocks such as camera movements, visual highlights, and animated elements. Our system includes a researcher component that accurately queries geospatial information by leveraging an LLM with web search, enabling the automatic extraction of relevant regions, paths, and coordinates while allowing users to edit and query for changes or additional information to refine the results. Additionally, users can fine-tune parameters of these blocks through an interactive timeline editor. We detail the system's design and architecture, informed by formative interviews with professional animators and an analysis of 200 existing map animation videos. Our evaluation, which includes expert interviews (N=5) and a usability study (N=12), demonstrates that MapStory enables users to create map animations with ease, facilitates faster iteration, encourages creative exploration, and lowers barriers to creating map-centric stories.