Some of the twitter videos or accounts are private and some videos are geographically restricted by the owner. We cannot access these twitter videos, thus they cannot be downloaded.
When you left click on the download link it is normal the browser starts to play twitter video. You need to right click on download link and choose "save as" to save the twitter video.
On a PC, they are usually saved under "Downloads" folder. On IOS and android devices you can save twitter videos to browser download folder or directly to Photos app.
Most of the twitter videos we serve are in HD or Full HD format. This means that you can download highest quality twitter videos including 1080p twitter videos and 720p twitter videos.