The New Relic browser agent is a tool that helps monitor the performance, errors, and user experience of web applications and sites by collecting data on page load time, Ajax requests, JavaScript errors, and custom events, which can also be used to observe real user monitoring (RUM) data in web extension views. To use it in a web extension view, developers need to follow specific steps, including installing the browser agent as an NPM package and loading it into their extension view code. Once installed, the agent can detect RUM data and provide insights for optimizing and troubleshooting the web extension's performance and user experience.