From change to generated website how does it work?

You have created pull request.

If you create pull request, website will not be generated until your pull request is merged. After your pull request is merged, generation will work just like when you commit your changes to master branch.

Your pull request was accepted or you have pushed your changes to master.

1. You've pushed commit to master branch of getting-started repository.
2. Your push triggers GitHub webhook to Travis CI and starts the build (you can read more about it here)
3. Travis CI runs Raketask from Rakefile (rake site:deploy) that sets git configuration, builds website (bundle exec jekyll build) and commits generated website after your changes to gh-pages branch.

If something went wrong you will receive email that build failed and on https://travis-ci.org/timvideos/getting-started you will see build history.