Verifying that your cookbooks actually work when converging a node is essential. Only when you know that you can rely on your cookbooks, are you ready to run them anytime on your production servers.
Test Kitchen is Chef's integration testing framework. It enables you to write tests, which run after a VM is instantiated and converged, using your cookbook. Your tests run in that VM and can verify that everything works as expected.
This is in contrast to ChefSpec, which only simulates a Chef run. Test Kitchen boots up a real node and runs Chef on it. Your InSpec tests run by Test Kitchen see the real thing.
Let's see how you can write such integration tests for your cookbooks.