The gist of snapshot testing is asserting that a set of data hasn’t changed compared to a previous version, which is a snapshot of the data, to prevent regressions. The difference between a classic assertEquals and an assertMatchesSnapshot is that you don't write the expectation yourself when snapshot testing. When a snapshot assertion happens for the first time, it creates a snapshot file with the actual output, and marks the test as incomplete. Every subsequent run will compare the output with the existing snapshot file to check for regressions.
Snapshot testing is most useful larger datasets that can change over time, like serializing an object for an XML export or a JSON API endpoint.
He starts with a basic example, checking to verify that a string hasn't changed between test runs. The first run creates the snapshot (marking the test as incomplete) and then following runs verify - one passing (no change) and one failing (changing of the return string). He also includes examples of tests on objects, mentions where the snapshot files are stored and how to add drivers for additional snapshot storage types.