I contact the developer, Florian Schimanke, who explained the steps the application takes. “[In iOS 7], it is possible to include fonts in configuration profiles. You can do this for example using the Apple Configurator from the Mac App Store,” he wrote.
“[AnyFont] takes the fonts that are added to the app’s storage by the user via iTunes file sharing or the ‘Open in…’ dialog and creates a configuration profile from it so it can be installed on the device. AnyFont hands over the newly created profile to Safari which then takes the user to the installation process. When finished, the user is then taken back to AnyFont.”
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