The bin/internal/engine.version file controls which version of the Flutter engine to use. The file contains the commit hash of a commit in the https://github.com/flutter/engine repository. That hash must have successfully been compiled on https://build.chromium.org/p/client.flutter/ and had its artifacts (the binaries that run on Android and iOS, the compiler, etc) successfully uploaded to Google Cloud Storage.
The /bin/internal/engine.merge_method file controls how we merge a pull request created by the engine auto-roller. If it‘s squash, there’s only one commit for a pull request no matter how many engine commits there are inside that pull request. If it's rebase, the number of commits in the framework is equal to the number of engine commits in the pull request. The latter method makes it easier to detect regressions but costs more test resources.