1) ensure you have published the webapp.
2) Get the published url and add any parameter at the end of the url and hit return
3) this triggers the doGet(e) function which will log the event (e) along with any parameters.
4) don't run it manually after entering the url as you will simply overwrite the log file.
5) open a new browser tab and go to the apps script dashboard/
my executions to view the log. You should be able to see both manual and "doGet" executions.
6) just use console.log as logger.log is mainly for the old IDE.
I'm happy to help. I too used this set of videos to get started with apps script web apps, and I support him on Patreon.