How can i prove that the activation link sent to subscriber (anonymous user) has actually been activated?
You can check the Status of your subscribers by going to People > Newsletter subscriptions (/admin/people/simplenews). 



There are 4 places within Drupal admin navigation that Simplenews entries are interspersed into: 


Configuration » Web services » Newsletters (/admin/config/services/simplenews) for creating different Newsletters (you want at least one) 
Structure » Content types » Simplenews newsletter (/admin/structure/types/manage/simplenews) to choose and edit a Content type to use for the newsletter content
People » Newsletter subscriptions (/admin/people/simplenews) to list and filter all your Subscribers
Content » Newsletters (/admin/content/simplenews) to see all the created Newsletter issues and their status