Visit the link, Cmd-Option-J in Chrome to open DevTools.

Execute and capture result of $('#playerContainer').data().config.video_url

Take that link and enter it in https://cloudconvert.com/ and select “Select Files” dropdown, enter url.

Or, use a commandline tool:

Install pre-requisites

pip install cloudconvert requests

Download script: https://gist.github.com/c83c3e91ee3f9df21686bb50b4fbf904

Make it executable: chmod +x twitter-gif

Run it: twitter-gif TWEET-LINK outputfilename-optional