(I didn't fully re-watch this video, but these notes are from my observations after skipping through the video and making educated guesses)
From what I re-watched of this video it looks like Rich is evaluating the code in a running repl. This allows him to change the code, evaluate it, and see different behavior.
Many editors have support for evaluating code in a buffer in a Clojure repl. Here is some documentation on using CIDER with Emacs to get to interactively play with your code.
Unrelated to the video in question, if you are using ring for web development. You can use the wrap-reload middleware to have your code automatically reloaded when a file has changed and a request hits your web app. This is extremely useful when developing a Clojure web application.
For automatically reloading and running your clojure.tests I recommend lein-test-refresh. It is a Leiningen plug-in that monitors your project for file changes and when something changes it reloads and runs your tests. If you have tests for your project this greatly speeds up development.