There are a few methods to save videos from the Twitter app or website. First, and one the easiest and simplest ways, is to use a website downloader app like the one provided by savevideofast.com, navigate to savevideofast, paste your tweet URL, and hit the download button.
The second way is to use an MP4 extension on your web browser, like the Twitter video downloader. This extension allows you to convert Twitter videos to mp4 format directly. Install the extension in the Google Chrome web browser and run the extension while playing your Twitter video on the desktop/laptop. Your download will be saved automatically to your default download location.