Neon's API versioning allows developers to create multiple versions of an API with different database schemas, enabling gradual rollout of new features while keeping existing APIs functional. By using Neon's branching feature, developers can dynamically manage database versions and serve different datasets based on the API version. The guide demonstrates how to implement this project by creating a FastAPI backend service that connects to different database branches using Neon's API, and provides step-by-step instructions for setting up the project, connecting API versions to correct database branches, and testing created branches. With Neon's serverless Postgres platform, developers can ship faster via instant provisioning, autoscaling, and database branching.