Member and Developer tier accounts on App.net will now have a "bucket" of 10GB. This 10GB can be accessed by App.net applications in order to read/write files. Think about this storage space as space on your harddrive, or space on your iPod. Free tier accounts come with 500MB of storage.
It is important to remember that file storage is a backend feature that we make available to the App.net developer community. Every app / client may or may not take advantage of it in the same way or at all.
Our native client Alpha, demonstrates this file storage by adding image uploading to posts and there is a 3rd party app called ADN File Manager that allows you to manage your "bucket"
For more context on App.net file storage read our blog post announcing it.