The api cannot be accessed using a Web browser. You should make requests via another application or command line tool like curl. An example PHP implementation is available here.
Looks like the API page in the manual needs updating since 3.4.1, as the access url was updated: https://www.phplist.org/newslist/phplist-3-4-1-release-notes/. The manual should be updated today. You should use http://jodywhitesides.com/lists/api/v2/sessions (or https).