Saving a movie is not downloading it
The Save to My List button stores a reference to the title so you can find it again. It does not place a video file on your phone. Similarly, the list backup is a small text record of your choices and checkpoints, not an offline movie library.
Taking a film on a trip
Open your chosen viewing service before leaving and check whether that edition offers offline downloads on your device. Where downloading is supported, the service controls its storage, expiry and playback rules. A browser movie page alone does not supply an offline copy.
Choose the audio and subtitle edition you need before starting the download. Test it inside the viewing app while you still have a connection. That is especially useful when travelling with a tablet that will later have no mobile data.
Catflix app download searches
If you meant a shortcut to the website, use your browser's bookmark or home-screen option. This Catflix site does not provide an APK installer or movie-file download. The app page explains mobile access without confusing a website icon with an Android package.
Keep enough storage for the app and the selected download, and avoid clearing its data until you have finished watching. For list-only portability, use My List instead.