When you type "download Twitter videos online" in a search engine you will find numerous online download services for sites like Twitter, YouTube and Metacafe video download, for example, VideoGrabby YouTube Downloader. They are used in a similar way -copying the video URL and then pasting it to the online downloader. It seems pretty simple but you should pay attention as they are all online services, the video is usually firstly downloaded to the server, and then to your own computer and in this way the download speed can dramatically fluctuate according to the net speed.
Method 3: How to Save a Twitter Video with Browser Extensions/Add-ons
You can also download add-ons from the search engine to download a Twitter video.
Google: https://chrome.google.com/webstore/category/extensions
Firefox: https://addons.mozilla.org/en-US/firefox/
In the URLs above, search "download Twitter videos" and then there will be many related results come out. After installation, restart the browser, right-click on the video you want to download or find the download option near the video window.