The application checks for updates. It downloads the http://www.gmail-backup.com/files/version.txt file, which contains following information: last revision (SVN), last revision (version number), URI for download of new version. For downloading the file the standard Python library urllib2 is used. You can check the yesterday access log of HTTP server which I made public here.
The reason why it checks the new version after the start of backup is that we share a lot of lines of code between command-line version and GUI version and we want to check new available version on one place.
You can find more details at http://www.gmail-backup.com/why-does-app-connect-home.