FastAPI is a Python web framework that allows developers to quickly spin up REST services, but it can experience slowdowns due to various reasons such as heavy request payloads, blocking I/O operations, insufficient hardware resources, and outdated operating systems. To troubleshoot performance issues, developers can use New Relic's observability platform, which provides metrics, events, logs, and traces to gain deeper insights into the application's performance. By using New Relic, developers can identify the root cause of slowdowns and apply solutions such as making functions asynchronous or resizing images to improve performance. Additionally, New Relic offers a quickstart guide for installing the Python agent in FastAPI apps, which provides pre-built tools and a dashboard to monitor application performance.