Searchonymous helps to keep your Google searches private: When installed no tracking cookies are sent to Google while searching. That way Google will not be able to connect searches with your account or create a search history. Also you will not receive customized search results.
When the add-on is active it seems to Google that are logged out of your Google account during searches (see the the “Sign in” button at the top right). However you stay logged in on other Google services such as Youtube or Gmail.
If you want to change your Google search settings, make sure to allow the “Search Preferences Cookie” in the add-on settings. By default the preferences-cookie will be blocked which effectively discards any changes to your search settings.
Please note that there are other means of tracking this add-on can’t protect you against, such as using your IP address or browser fingerprint.