When you tap an image file, it is opened within the MOVEit app. When you tap other types of files, the file is opened in an application on your device. You will be prompted to select an app if more than one app on your device is registered to open files of that file type. (If you have no registered apps for the file type, you will receive a dialog saying the file cannot be opened.)
Files are only temporarily saved on the device while you view them. The originals remain saved on the MOVEit server, either attached to packages (which are listed in mailboxes) or in folders.
Note: Your ability to save a file as a local copy on your device depends on the app you use. The MOVEit mobile app itself does NOT offer a "Download" function or "Save as..." dialog to let you save a file to local device storage.