Thereafter, this secret key is used by both sender and receiver to encrypt the current time and produce a hash code.
If the codes match and your login credentials are valid (2 factors), you are authenticated; otherwise, you are not.
Once you dig into it, it's not too hard to replicate. I wrote myself a simple desktop CLI program that duplicates the app on my phone. It retrieves the appropriate secret key (selected by key/name) from an encrypted data file and generates a code as needed and places it into the clipboard so I can easily paste into a login screen.
Also noteworthy: the current time is divided into coarse time periods (typically UNIX time divided by 30s?), meaning that slight differences in clock time of all participants don't matter because they can reproduce the same hash code during the time period.
Time increment rollover is an issue even if clocks are synced perfectly. This is due to an uncontrollable human factor --- delay from the time the code is generated until it is input and submitted.
To account for this, most receivers will check the provided hash code against both the current and the previous time increment.