That is false; the license doesn't speak about any such right.
Rather, the copyright holder of the AGPLed work is exercising their right to dictate the following:
[Y]our modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software.
Under the AGPL, the visitors to the program need not request the source; it must be prominently available for them to obtain in a self-serve manner. No mention is made that they have any kind of right. They are not parties to the license at all. A copyright license can only grant rights to licensees.
By the way, "all users interacting" could be interpreted to literally mean all users, including users who visit the login page, but do not have an account (and are not authorized to have one due to not belonging to the organization that runs the service).
Rather, the copyright holder of the AGPLed work is exercising their right to dictate the following:
[Y]our modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software.
Under the AGPL, the visitors to the program need not request the source; it must be prominently available for them to obtain in a self-serve manner. No mention is made that they have any kind of right. They are not parties to the license at all. A copyright license can only grant rights to licensees.
By the way, "all users interacting" could be interpreted to literally mean all users, including users who visit the login page, but do not have an account (and are not authorized to have one due to not belonging to the organization that runs the service).